Combined optical metrology techniques

ABSTRACT

A method and apparatus is disclosed for using below deep ultra-violet (DUV) wavelength reflectometry for measuring properties of diffracting and/or scattering structures on semiconductor work-pieces is disclosed. The system can use polarized light in any incidence configuration, but one technique disclosed herein advantageously uses un-polarized light in a normal incidence configuration. The system thus provides enhanced optical measurement capabilities using below deep ultra-violet (DUV) radiation, while maintaining a small optical module that is easily integrated into other process tools. A further refinement utilizes an r-θ stage to further reduce the footprint.

CROSS-REFERENCE TO RELATED APPLICATION

This application is a divisional of U.S. patent application Ser. No.12/080,947, filed Apr. 7, 2008, which claims priority to U.S.Provisional Patent Application 60/922,434, filed Apr. 9, 2007, entitled“Method And System For Using Reflectometry Below Deep Ultra-Violet (DUV)Wavelengths For Measuring Properties Of Diffracting Or ScatteringStructures On Substrate Work-Pieces.”

TECHNICAL FIELD OF THE INVENTION

This disclosure relates to a method for using single-wavelength,multiple-wavelength, or broadband reflectometry that include below deepultra-violet (DUV) wavelengths for measuring properties of diffractingand/or scattering structures on substrate work-pieces, such as, forexample, semiconductor substrates.

BACKGROUND

Optical methods for control of critical dimensions and/or profile ofetched and lithographic structures in high-volume semiconductormanufacturing environments are gaining wide acceptance, largely due tothe promise of rapid, nondestructive real-time feedback forcost-effective process control.

Among the earliest current art metrology systems are scatterometrysystems, such as the method taught in U.S. Pat. Nos. 5,164,790 or5,703,692, which determine angle-resolved spectral response fromperiodic structures. Later current art metrology systems employedtraditional thin film analysis tools, such as broad-band reflectometersand ellipsometers, as taught in U.S. Pat. Nos. 5,432,607, 6,281,674, or6,898,537.

Most of the various current designs operate in a spectral region betweendeep ultraviolet (DUV) (˜200 nm) and near infra-red (˜1000 nm)wavelengths. This limits the fundamental resolution of such systems whenmeasuring structures much smaller than the incident wavelength, andcauses the metrology to lose sensitivity to the details of profileshape. As such, current optical metrology becomes increasingly obsoleteas semiconductor device dimensions shrink.

At a given wavelength range, the more incident conditions an opticaltool measures, the greater the sensitivity of the measurement to agreater number of parameters. Accordingly, some recent current artsystems overcome some of the resolution issue by combining ellipsometricand polarimetric (polarized reflectance) data, such as the methodstaught in U.S. Pat. Nos. 6,713,753 and 6,590,656, at the expense ofgreater complexity and less versatility in a manufacturing environment.Another approach combines broadband reflectance, polarimetric, orellipsometric data with multiple angle of incidence measurements, suchas the method taught in the article T. Novikova, A. Martino, S. B.Hatit, and B. Drevillon, “Application of Mueller polarimetry in conicaldiffraction for critical dimension measurements in microelectronics”,Appl. Opt., Vol. 45, No. 16, p. 2006. Such systems are complicated tooperate, often slow, and are very hard to integrate into themanufacturing process. Aside from this, there is still the fundamentalissue that resolution information is lost as the measured feature sizesdecrease, and after a certain point, no amount of additional datasetswill compensate for this.

On another front, optical data from metrology tools are often analyzedusing rigorous solutions to the boundary value problem. One of the mostcommon analysis technique for periodic structures is the rigorouscoupled wave (RCW) method, which is sometimes also referred to as theFourier Modal method. The RCW method is used to compute theoreticaloptical spectra representative of the structure being measured as themodel parameters are changed during a regression analysis. The optimizedparameters are the measurement result.

The RCW calculation can be very computationally intensive. In somecases, a library database is used to store pre-generated spectra to becompared with the measured spectra during measurement. Even then, theefficiency of the calculation is important since hundreds of thousandsor even millions of spectra can be required for the database.

The special case of normal incidence benefits from symmetry conditionsat all wavelength ranges, allowing for the most efficient RCWcalculations. In addition, a normal incidence reflectometer is moresuited to integration into the device manufacturing process, being lesscomplicated to operate, easier to maintain, and more compact than theangle-resolved or ellipsometric solutions mentioned above.

Thus, it is desirable to have a reflectometer configured for normalincidence measurement for practical reasons, but also capable of usingbelow deep ultra-violet (DUV) wavelength light for enhanced measurementcapabilities. Instances of normal incidence polarized reflectometry inthe current art, such as the one disclosed in U.S. Pat. No. 6,898,537,are not suitable for operation below DUV. The patent teaches acalibration method to account for the offset between differentpolarization conditions, which will not work in the region below DUV dueto contaminant buildup during the tool's operation. In general, it isquite difficult to polarize light below ˜160 nm In addition, thecalibration of the absolute reflectance used by the system disclosed inU.S. Pat. No. 6,898,537 is complicated by the lack of reliablereflectance reference standards in the range below DUV. Therefore, themethod disclosed in U.S. Pat. No. 6,898,537 is unsuitable for work belowDUV wavelength range. A further complication arises in the use ofpolarized reflectance with an r-O stage, and an elaborate polarizationalignment procedure is required during measurement of periodicstructures, since the orientation of the structures will vary as afunction of r-θ position.

SUMMARY

The techniques disclosed herein measure broadband below deepultra-violet DUV-Visible (Vis) or near infra-red (NIR) reflectancespectra from diffracting and scattering features. One technique of thesystem uses a wavelength range of 120 nm-800 nm The wide wavelengthrange provides a large set of incident conditions for improvedsensitivity to multiple parameters, negating the need for complicatedarrangements to impose multiple angle and polarization conditions. Inaddition, the inclusion of the portion of the spectrum below DUVenhances sensitivity to smaller feature sizes. The techniques disclosedherein also use an r-θ stage together with un-polarized normal incidencereflectance so that a smaller footprint is retained without the need forcomplicated polarization alignment. In addition, a faster calculationspeed can be achieved for periodic structures by exploiting the naturalsymmetry of the diffraction calculation in a normal incident condition.

The present disclosure provides a method of optically measuringdiffracting and scattering structures on a sample, comprising providinga below deep ultra-violet (DUV)—Vis referencing reflectometer, whereinreferencing is used to account for system and environmental changes toadjust reflectance data obtained through use of the reflectometer,providing at least one computer, and extracting structural and opticalparameters from a theoretical model of the diffracting and scatteringstructure via a computer.

In one embodiment, the referencing reflectometer is configured fornormal incidence, allowing for use of a reduced RCW calculation whenanalyzing 2-D periodic structures, or use of a group theoretic approachwhen analyzing 3-D periodic structures, to take advantage of thesymmetry. It should be pointed out that while a reduced RCW calculationis advantageous, its use is not required. Use of the full RCWcalculation as well as analysis methods other than RCW, which may or maynot make use of symmetry, is not precluded. The system can also be usedto measure non-symmetric periodic structures (using, e.g. the full RCWor other rigorous method) as well as non-periodic structures, employingany number of methods available in the literature, either rigorous orapproximate. The incident light can be un-polarized.

In one embodiment, a reflectometer apparatus for analyzing a scatteringor diffracting structure is provided. The reflectometer may comprise abelow deep ultra-violet (DUV) wavelength referencing reflectometerconfigured for normal incidence operation and having a light source thatprovides at least below DUV wavelength light, wherein referencing isconfigured to account for system and environmental changes to adjustreflectance data obtained through use of the reflectometer. Thereflectometer may also comprise at least one computer connected to thereflectometer and a computer program for use with the at least onecomputer configured to extract structural and optical parameters from atheoretical model of the scattering or diffracting structure. Thecomputer program uses a reduced RCW calculation for analyzing 2-Dperiodic structures of the scattering or diffracting structure.

In one embodiment, a method of optically measuring diffracting andscattering features on a sample is disclosed. The method may compriseproviding an optical signal having at least some below deep ultravioletlight wavelengths and directing the light on the sample in asubstantially normally incident configuration, wherein the incidentlight is un-polarized. The method may further comprise utilizing areduced RCW calculation to analyze 2-D periodic structures and utilizinga group theoretic approach to analyze 3-D periodic structures.

In another embodiment a method of optically measuring diffracting andscattering features on a sample is disclosed. The method may compriseproviding a reflectometer that utilizes at least some below deepultra-violet wavelengths of light and measuring intensity data from aplurality of sites within an area of the sample. The method may furthercomprise analyzing a combination of the measured intensity data from theplurality of sites that is independent of incident intensity in order toextract structural and/or optical property information regarding thesample.

In another embodiment a method of optically measuring diffracting andscattering features on a sample is disclosed. The method may compriseproviding a reflectometer that utilizes at least some below deepultra-violet wavelengths of light and measuring intensity data from aplurality of sites within an area of the sample. At least one of thesites represents an unpatterned region of the sample and at least oneother site represents a patterned region of the sample.

In another embodiment, a method for measuring properties of a sample isdisclosed. The method comprises providing an optical metrology tool thatincludes a first optical metrology apparatus, the first opticalmetrology apparatus being a first reflectometer having at least in partbelow deep ultra-violet light wavelengths, and providing a secondoptical metrology apparatus within the optical metrology tool, thesecond optical metrology apparatus providing optical measurements forthe sample utilizing a different optical metrology technique as comparedto the first optical metrology apparatus. Data sets from the firstoptical metrology apparatus and the second optical metrology apparatusare combined and analyzed in order to measure at least one property ofthe sample.

In another embodiment a reflectometer apparatus for analyzing ascattering or diffracting structure is disclosed. The apparatus maycomprise a below deep ultra-violet (DUV) wavelength referencingreflectometer configured for normal incidence operation and having anunpolarized light source and non-polarizing optical system that providesat least below deep ultra-violet wavelength light, wherein referencingis configured to account for system and environmental changes to adjustreflectance data obtained through use of the reflectometer. Theapparatus may further comprise at least one computer connected to thereflectometer, and a computer program for use with the at least onecomputer configured to extract structural and optical parameters from atheoretical model of the scattering or diffracting structure. Theapparatus may further comprise an r-θ stage for holding the scatteringor diffracting structure, wherein a calculated reflectance is obtainedfrom a relationship that is independent of a sample rotation.

As described below, other features and variations can be implemented, ifdesired, and a related method can be utilized, as well.

DESCRIPTION OF THE DRAWINGS

It is noted that the appended drawings illustrate only exemplaryembodiments of the invention and are, therefore, not to be consideredlimiting of its scope, for the invention may admit to other equallyeffective embodiments.

FIG. 1 is a schematic representation of a reflectometer.

FIG. 2 is a more detailed schematic representation of a reflectometer.

FIG. 3 is schematic illustrating polar (theta) and azimuth (phi)incident angles.

DETAILED DESCRIPTION OF EMBODIMENTS

The techniques disclosed herein involve an extension of the recenttechnology taught in U.S. Pat. No. 7,067,818 titled “Vacuum UltravioletReflectometer System and Method”, U.S. Pat. No. 7,026,626 titled“Semiconductor Processing Techniques Utilizing Vacuum UltravioletReflectometer”, and U.S. Pat. No. 7,126,131 titled “Broad BandReferencing Reflectometer”, which are all expressly incorporated intheir entirety herein by reference. One technique measures reflectancespectrum in the 120 nm-800 nm wavelength range, providing a much greaterspectral range than any existing reflectometer. Preferred techniquesdisclosed herein operate at normal incidence partly to minimize theoverall footprint. Another technique additionally uses an r-θ stage,further reducing the footprint of the sample area.

To enhance the sensitivity of optical metrology equipment forchallenging applications it is desirable to extend the range ofwavelengths over which such measurements are performed. Specifically, itis advantageous to utilize shorter wavelength (higher energy) photonsextending into, and beyond, the region of the electromagnetic spectrumreferred to as the vacuum ultra-violet (VUV). Historically there hasbeen relatively little effort expended on the development of opticalinstrumentation designed to operate at these wavelengths, owing to thefact that VUV (and lower) photons are strongly absorbed in standardatmospheric conditions. Vacuum ultra-violet (VUV) wavelengths aregenerally considered to be wavelengths less than deep ultra-violet (DUV)wavelengths. Thus VUV wavelengths are generally considered to bewavelengths less than about 190 nm. While there is no universal cutofffor the bottom end of the VUV range, some in the field may consider VUVto terminate and an extreme ultra-violet (EUV) range to begin (forexample some may define wavelengths less than 100 nm as EUV). Though theprinciples described herein may be applicable to wavelengths above 100nm, such principles are generally also applicable to wavelengths below100 nm Thus, as used herein it will be recognized that the term VUV ismeant to indicate wavelengths generally less than about 190 nm howeverVUV is not meant to exclude lower wavelengths. Thus as described hereinVUV is generally meant to encompass wavelengths generally less thanabout 190 nm without a low end wavelength exclusion. Furthermore, lowend VUV may be construed generally as wavelengths below about 140 nm.

Indeed it is generally true that virtually all forms of matter (solids,liquids and gases) exhibit increasingly strong optical absorptioncharacteristics at VUV wavelengths. Ironically it is this same ratherfundamental property of matter which is partly (along with decreasedwavelength versus feature size) responsible for the increasedsensitivity available to VUV optical metrology techniques. This followsas small changes in process conditions, producing undetectable changesin the optical behavior of materials at longer wavelengths, can inducesubstantial and easily detectable changes in the measurablecharacteristics of such materials at VUV wavelengths.

The fact that VUV photons are strongly absorbed by most forms of matterprecludes the simple extension of, or modification to, conventionallonger wavelength optical metrology equipment in order to facilitateoperation in the VUV. Current day tools are designed to operate understandard atmospheric conditions and typically lack, among other things,the controlled environment required for operation at these shorterwavelengths. VUV radiation is strongly absorbed by both O₂ and H₂Omolecules and hence these species must be maintained at sufficiently lowlevels as to permit transmission of VUV photons through the optical pathof the instrument. The transmission of photons through standardatmosphere drops precipitously at wavelengths shorter than about 200 nm.

Not only are conventional optical instruments intended to function instandard atmospheric conditions, they also typically employ an array ofoptical elements and designs which render them unsuitable for VUVoperation. In order to achieve highly repeatable results with areflectometer it is desirable to provide a means by which reflectancedata can be referenced or compared to a relative standard. In thismanner changes in the system that occur between an initial time when thesystem is first calibrated and a later time when a sample measurement isperformed, can be properly accounted for. At longer wavelengths suchchanges are usually dominated by intensity variations in the spectraloutput of the source. When working at VUV wavelengths, however, changesin the environmental conditions (i.e. changes in the concentration ofabsorbing species in the environment of the optical path) can play amuch larger role.

Thus, conventional longer wavelength systems fail to address thesignificant influence that the absorbing environment has on themeasurement process. To ensure that accurate and repeatable reflectancedata is obtained, it is desirable to not only provide a means ofcontrolling the environment containing the optical path, but furthermoreto ensure that the absorption effects which do occur are properly takeninto account during all aspects of the calibration, measurement andreference processes.

Hence, it is desirable to provide an optical metrology tool with acontrolled environment that is designed to operate at and below VUVwavelengths. In addition, in order to ensure that accurate andrepeatable results are obtained, it is desirable that the designincorporate a robust referencing methodology that acts to reduce oraltogether remove errors introduced by changes in the controlledenvironment.

Examples of a VUV optical metrology instrument well suited to benefitfrom use of the methods herein described are disclosed in U.S.application Ser. No. 10/668,642, filed on Sep. 23, 2003, now U.S. Pat.No. 7,067,818; U.S. application Ser. No. 10/909,126, filed on Jul. 30,2004, now U.S. Pat. No. 7,126,131; and U.S. application Ser. No.11/600,413, filed on Nov. 16, 2006 now U.S. Pat. No. 7,342,235, thedisclosures of which are all expressly incorporated in their entiretyherein by reference. The metrology instrument may be a broad-bandreflectometer specifically designed to operate over a broad range ofwavelengths, including the VUV. A schematic representation of an opticalreflectometer metrology tool 1200 that depicts one technique disclosedherein is presented in FIG. 1. As is evident, the source 1210, beamconditioning module 1220, optics (not shown), spectrometer 1230 anddetector 1240 are contained within an environmentally controlledinstrument (or optics) chamber 1202. The sample 1250, additional optics1260, motorized stage/sample chuck 1270 (with optional integrateddesorber capabilities) and sample are housed in a separateenvironmentally controlled sample chamber 1204 so as to enable theloading and unloading of samples without contaminating the quality ofthe instrument chamber environment. The instrument and sample chambersare connected via a controllable coupling mechanism 1206 which canpermit the transfer of photons, and if so desired the exchange of gasesto occur. A purge and/or vacuum system 1280 may be coupled to theinstrument chamber 1202 and the sample chamber 1204 such thatenvironmental control may be exercised in each chamber.

Additionally a computer 1290 located outside the controlled environmentmay be used to analyze the measured data. A computer program forextracting structural and optical parameters from a theoretical model ofthe diffracting and scattering structure is included in the computer1290. The referencing reflectometer is configured for normal incidence.A reduced RCW calculation can be used for analyzing 2-D periodicstructures to take advantage of the symmetry. Similarly, a grouptheoretic approach can be used for analyzing 3-D periodic structures totake advantage of the symmetry. The incident light can be un-polarized.It will be recognized that computer 1290 may be any of a wide variety ofcomputing or processing means that may provide suitable data processingand/or storage of the data collected.

While not explicitly shown in FIG. 1, it is noted that the system couldalso be equipped with a robot and other associated mechanized componentsto aid in the loading and unloading of samples in an automated fashion,thereby further increasing measurement throughput. Further, as is knownin the art load lock chambers may also be utilized in conjunction withthe sample chamber to improve environmental control and increase thesystem throughput for interchanging samples.

In operation light from the source 1210 is modified, by way of beamconditioning module 1220, and directed via delivery optics through thecoupling mechanism windows 1206 and into the sample chamber 1204, whereit is focused onto the sample by focusing optics 1260. Light reflectedfrom the sample is collected by the focusing optics 1260 and re-directedout through the coupling mechanism 1206 where it is dispersed by thespectrometer 1230 and recorded by a detector 1240. The entire opticalpath of the device is maintained within controlled environments whichfunction to remove absorbing species and permit transmission of belowDUV photons.

Referring again to FIG. 1, the beam conditioner module 1220 allows forthe introduction of spatial and/or spectral filtering elements to modifythe properties of the source beam. While this functionality may notgenerally be required, there may arise specific applications where it isdeemed advantageous. Examples could include modifying the spatial ortemporal coherence of the source beam through use of an aperture, orintroduction of a “solar blind” filter to prevent longer wavelengthlight from generating spurious below DUV signals through scatteringmechanisms that may occur at the various optical surfaces in the opticalbeam path.

The beam conditioner can also include a polarizer, which would be usefulfor critical dimension measurements where it is desirable to polarizethe incident light in a particular direction with respect to themeasured structures. Alternately, it may be desirable to have anon-polarizing optical path, and the beam conditioner can consist of adepolarizer to counter the effects of any polarization imparted by thepreceding optics. Additionally, either a polarizing or depolarizing beamconditioner can be placed in the optical path on the detection side ofthe sample. A depolarizer at this location would be useful foreliminating any polarization effects of the detection system.

While in some techniques disclosed herein the reflectance data can bepolarized in particular directions with respect to a diffractingstructure, one technique uses an un-polarized broadband source. This ispartly advantageous due to the difficulty in polarizing below deepultra-violet (DUV) light, but also allows a more straight-forward use ofan r-θ stage, since the normal incidence un-polarized spectrum is thesame regardless of sample orientation. This technique is advantageous inhigh volume manufacturing environments, and in particular is well-suitedto integrated applications.

These advantages are retained without giving up measurement capabilityof the system. The below DUV portion of the spectra is potentially muchricher than DUV-visible (DUV-Vis) light, for both scattering andnon-scattering structures, for two primary reasons: 1) the wavelengthvs. feature size is much smaller than with conventional DUV-Vis opticalmetrology, and 2) many materials that have relatively featurelessdispersions in the DUV-Vis range have very rich absorption spectra inthe below DUV range, which leads to a stronger response of the spectraat these wavelengths. In combination, the inclusion of the below DUVspectrum can easily make up for or exceed the additional spectralinformation contained in conventional DUV-Vis multiple angleellipsometric configurations.

An additional difficulty in using below DUV spectrometry is caused by acontaminant buildup that occurs on optical components and referencesamples due to the interaction of common fab materials with high energyradiation. This contaminant buildup has particular relevance to absolutereflectance calibration, since it is difficult to maintain a consistentreference sample. Accordingly, one technique disclosed hereinincorporates new calibration procedures as described in U.S. applicationSer. No. 10/930,339, filed on Aug. 31, 2004, and also described in U.S.application Ser. No. 11/418,827, filed on May 5, 2006, now U.S. Pat. No.7,282,703, and U.S. application Ser. No. 11/418,846, filed on May 5,2006, all of which are incorporated herein in their entirety byreference.

A more detailed schematic of the optical aspects of the instrument ispresented in FIG. 2. The instrument is configured to collect referencedbroad band reflectance data in the below DUV and two additional spectralregions. In operation light from these three spectral regions may beobtained in either a parallel or serial manner. When operated in aserial fashion reflectance data from the below DUV is first obtained andreferenced, following which, reflectance data from the second and thenthird regions is collected and referenced. Once all three data sets arerecorded they are spliced together to form a single broad band spectrum.In parallel operation reflectance data from all three regions arecollected, referenced and recorded simultaneously prior to datasplicing.

The instrument is separated into two environmentally controlledchambers, the instrument chamber 2102 and the sample chamber 2104. Theinstrument chamber 2102 houses most of the system optics and is notexposed to the atmosphere on a regular basis. The sample chamber 2104houses the sample and reference optics, and is opened regularly tofacilitate changing samples. For example, the instrument chamber 2102may include mirrors M-1, M-2, M-3, and M-4. Flip-in mirrors FM-1 andFM-3 may be utilized to selective chose which light source 2201, 2202and 2203 is utilized (each having a different spectral region). Flip-inmirrors FM-2 and FM-4 may be utilized to selective chose one ofspectrometers 2204, 2216, and 2214 (again depending upon the chosenspectral region). As mentioned above with reference to FIG. 1, thespectrometers may be any of a wide variety of types of spectrometers.Mirrors M-6, M-7, M-8 and M-9 may be utilized to help direct the lightbeams as shown. Windows W-1 and W-2 couple light between the instrumentchamber 2102 and sample chamber 2104. Windows W-3, W-4, W-5 and W-6couple light into and out of the instrument chamber 2102. Beam splitterBS and shutters S-1 and S-2 are utilized to selectively direct light toa sample 2206 or a reference 2207 with the assistance of mirrors M-2 andM-4 as shown (the reference may be a mirror in one embodiment). Thesample beam passes through compensator plate CP. The compensator plateCP is included to eliminate the phase difference that would occurbetween the sample and reference paths resulting from the fact thatlight traveling in the sample channel passes through the beam splittersubstrate but once, while light traveling in the reference channelpasses through the beam splitter substrate three times due to the natureof operation of a beam splitter. Hence, the compensator plate may beconstructed of the same material and is of the same thickness as thebeam splitter. This ensures that light traveling through the samplechannel also passes through the same total thickness of beam splittersubstrate material.

When operated in a serial fashion below DUV data is first obtained byswitching the second spectral region flip-in source mirror FM-1 andthird spectral region flip-in source mirror FM-2 into the “out” positionso as to allow light from the below DUV source to be collected,collimated and redirected towards beam splitter element BS by thefocusing mirror M-1. Light striking the beam splitter is divided intotwo components, the sample beam 2255 and the reference beam 2265, usinga near-balanced Michelson interferometer arrangement. The sample beam isreflected from the beam splitter BS and travels through the compensatorplate CP, sample shutter S-1 and sample window W-1 into the samplechamber 2104, where it is redirected and focused onto the sample 2206via a focusing mirror M-2. The reference shutter S-2 is closed duringthis time. The sample window W-1 is constructed of a material that issufficiently transparent to below DUV wavelengths so as to maintain highoptical throughput.

Light reflected from the sample is collected, collimated and redirectedby the sample mirror M-2 back through the sample window, where it passesthrough the sample shutter and compensator plate. The light thencontinues on unhampered by the first spectral region flip-in detectormirror FM-2 and the second spectral region flip-in detector mirror FM-4(switched to the “out” position), where it is redirected and focusedonto the entrance slit of the below DUV spectrometer 2214 by thefocusing mirror M-3. At this point light from the sample beam isdispersed by the VUV spectrometer and recorded by its associateddetector. The spectrometer may be any of a wide variety of spectrometersincluding those types disclosed in U.S. application Ser. No. 11,711,482,filed on Feb. 27, 2007, the disclosure of which is incorporated in itsentirety herein. Thus, the spectrometer configuration is not intended tobe limited to the particular configuration shown in the figure.

Following collection of the sample beam, the reference beam is measured.This is accomplished by closing the sample shutter S-1 and opening thereference shutter S-2. This enables the reference beam to travel throughthe beam splitter BS, reference shutter S-2 and reference window W-2into the sample chamber 2104, wherein it is redirected and focused bymirror M-4 onto the plane reference mirror 2207 which serves as thereference. The reference window is also constructed of a material thatis sufficiently transparent to VUV wavelengths so as to maintain highoptical throughput.

Light reflected from the surface of the plane reference mirror 2207travels back towards the focusing reference mirror M-4 where it iscollected, collimated and redirected through the reference window W-2and the reference shutter S-2 towards the beam splitter BS. Light isthen reflected by the beam splitter towards the focusing mirror M-3where it is redirected and focused onto the entrance slit of the VUVspectrometer 2214. The path length of the reference beam 2265 isspecifically designed so as to match that of the sample beam 2255 ineach of the environmentally controlled chambers.

Following measurement of the below DUV data set, the second spectralregion data set is obtained in a similar manner During collection of thesecond region spectral data both the second spectral region sourceflip-in mirror FM-1 and the second spectral region detector flip-inmirror FM-2 are switched to the “in” position. As a result, light fromthe below DUV source 2201 is blocked and light from the second spectralregion source 2203 is allowed to pass through window W-3, after it iscollected, collimated and redirected by its focusing mirror M-6.Similarly, switching the second spectral region detector flip-in mirrorFM-2 into the “in” position directs light from the sample beam (when thesample shutter is open and the reference shutter is closed) andreference beam (when the reference shutter is open and the sampleshutter is closed) through the associated window W-6 and onto the mirrorM-9 which focuses the light onto the entrance slit of the secondspectral region spectrometer 2216, where it is dispersed and collectedby its detector.

Data from the third spectral region is collected in a similar fashion byflipping “in” the third spectral region source flip-in mirror FM-3 andthe third spectral region detector flip-in mirror FM-4, while flipping“out” the second spectral region source flip-in mirror FM-1 and thesecond spectral region detector flip-in mirror FM-2.

Once the sample and reference measurements for each of the spectralregions have been performed, a computer or processor (not shown) can beused to calculate the referenced reflectance spectra in each of thethree regions. Finally, these individual reflectance spectra arecombined to generate a single reflectance spectrum encompassing thethree spectral regions.

When operated in a parallel mode, the source and detector flip-inmirrors are replaced with appropriate beam splitters so that data fromall three spectral regions are recorded simultaneously.

Again, a polarizer can be included in the incident optical path beforethe beam splitter in order to polarize the incident light in aparticular direction with respect to the measured structures.Alternately, it may be desirable to have a non-polarizing optical path,and a non-polarizing beam splitter can be used in conjunction with anunpolarized source. If necessary, a depolarizer can be included in theoptical path just before the beam splitter to counter the effects of anypolarization imparted by the preceding optics. Additionally, either apolarizing or depolarizing beam conditioner can be placed in the opticalpath on the detection side of the sample, after the beam splitter. Adepolarizer at this location would be useful for eliminating anypolarization effects of the detection system.

The spectral information is generally analyzed using regression orlibrary techniques. The techniques disclosed herein may take advantageof the symmetry of the normal incidence configuration by reformulatingthe RCW eigen-problem for the normal incidence case. For two dimensionalperiodic structures, a method distinct from that taught in U.S. Pat. No.6,898,537 is described here, in that the current method uses a moregeneral derivation that leads to a different eigen-problem formulation,is more efficient in its treatment of the TM case, and does not requiremodification of the form of the boundary problem. For three dimensionalstructures, such as contact holes or vias, the formulation disclosed inthe techniques described herein for 2-D structures can be generalized.Alternately the methods in Benfeng Bai and Lifeng Li, “Reduction ofcomputation time for crossed-grating problems: a group theoreticapproach,” J. Opt. Soc. Am. A 21, 1886-1894 (2004), and subsequentpublications can be employed to take advantage of the normal incidenceconfiguration for 3-D periodic structures.

The steps leading to the reduced RCW computation for 2-D structures arenow described. The description follows the treatment and notationdescribed in M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K.Gaylord, “Formulation for stable and efficient implementation of therigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A12, 1068-1076 (1995) and illustrated in FIG. 3. Note that unreducedeigen-problem matrix and vector indices run from −N to N, with the (−N,−N) matrix element at the top left corner, in order to be consistentwith a symmetric diffraction problem with positive and negative orders.When creating a computer algorithm, we will need to label the indicesfrom 1 to 2N+1, or 0 to 2N, depending on the programming language used.Obviously, this is a notation preference and has no effect on theoutcome. The indices of the reduced matrices will run from 0 to N ineither case.

FIG. 3 illustrates the geometry of the problem. We first note that it ispossible to decouple two independent incident polarizations—TE and TM,as long as the plane of incidence is in the phi=0 configuration. Thatis, the plane of incidence is perpendicular to the grating lines. Anarbitrary polarization can be expressed as a linear combination of theTE and TM cases. In particular, for an un-polarized incident beam, whichwill include equal components of all possible polarizations, one cantake the average over all of the polarization conditions to obtain

$\begin{matrix}{{R = {\frac{1}{2}\left( {R_{TE} + R_{TM}} \right)}},} & {{eq}.\mspace{14mu} 1}\end{matrix}$

where R is the specular zero-order reflectance. This means that thereflectance can be computed for un-polarized incident light by computingthe TE and TM reflectances separately, and then taking the average. Theadvantage to doing this is that with the normal incidence condition, thegrating orientation is immaterial—the reflectance from the grating canalways be obtained from eq. 1 for un-polarized incident light.

First, the conventional formulation for the TE case must be described.In this case, the electric field has only a y-component 3100 (parallelto the grating lines), while the magnetic field H has both x-3102 andz-components 3104, but no y-component. The fields in each of the 3regions shown in FIG. 3 are expanded as generalized Fourier series:

$\begin{matrix}{{E_{{inc},y} = {\exp \left\lbrack {{- j}\; k_{0}{n_{I}\left( {{\sin \; \theta \; x} + {\cos \; \theta \; z}} \right)}} \right\rbrack}},} & {{eq}.\mspace{14mu} 2} \\{E_{I,y} = {E_{{inc},y} + {\sum\limits_{i = {- \infty}}^{\infty}{R_{i}{\exp \left\lbrack {- {j\left( {k_{xi} - {k_{I,{zi}}z}} \right)}} \right\rbrack}}}}} & {{eq}.\mspace{14mu} 3}\end{matrix}$

in the incident region (k₀=2π/2),

$\begin{matrix}{{E_{{II},y} = {\sum\limits_{i = {- \infty}}^{\infty}{T_{i}\exp \left\{ {- {j\left\lbrack {{k_{xi}x} + {k_{{II},{zi}}\left( {z - d} \right)}} \right\rbrack}} \right\}}}},} & {{eq}.\mspace{14mu} 4}\end{matrix}$

in the substrate medium, and

$\begin{matrix}{{E_{gy} = {\sum\limits_{i = {- \infty}}^{\infty}{{S_{yi}(z)}{\exp \left( {{- j}\; k_{xi}x} \right)}}}},} & {{eq}.\mspace{14mu} 5} \\{H_{gx} = {{- {j\left( \frac{ɛ_{f}}{\mu_{f}} \right)}^{1/2}}{\sum\limits_{i = {- \infty}}^{\infty}{{U_{xi}(z)}{\exp \left( {{- j}\; k_{xi}x} \right)}}}}} & {{eq}.\mspace{14mu} 6}\end{matrix}$

for the tangential fields in the grating region, where

$\begin{matrix}{{k_{xi} = {k_{0}\left\lbrack {{n_{I}\sin \; \theta} - {i\left( {\lambda_{0}/\Lambda} \right)}} \right\rbrack}},{and}} & {{eq}.\mspace{14mu} 7} \\{k_{l,{zi}} = \left\{ {\begin{matrix}{k_{0}\left\lbrack {n_{l}^{2} - \left( {k_{xi}/k_{0}} \right)^{2}} \right\rbrack}^{1/2} & {{k_{0}n_{l}} > k_{xi}} \\{- {{jk}_{0}\left\lbrack {\left( {k_{xi}/k_{0}} \right) - n_{l}^{2}} \right\rbrack}^{1/2}} & {k_{xi} > {k_{0}n_{l}}}\end{matrix},\mspace{14mu} {l = I},{II}} \right.} & {{eq}.\mspace{14mu} 8}\end{matrix}$

ε_(f) is the permittivity of free space, and μ_(f) is the magneticpermeability of free space. The permittivity in the grating region isalso expanded as a Fourier series:

$\begin{matrix}{{{ɛ(x)} = {\sum\limits_{h}{ɛ_{h}{\exp \left( {j\frac{2\pi \; h}{\Lambda}} \right)}}}},{ɛ_{0} = {{n_{rd}^{2}f} + {n_{gr}^{2}\left( {1 - f} \right)}}},{ɛ_{h} = {\left( {n_{rd}^{2} - n_{gr}^{2}} \right)\frac{\sin \left( {\pi \; {hf}} \right)}{\pi \; h}}},} & {{eq}.\mspace{14mu} 9}\end{matrix}$

where n_(rd) is the complex index of refraction of the grating ridges,and n_(gr) is the complex index of refraction of the grating groves.

The fields everywhere satisfy the Maxwell equation:

$\begin{matrix}{{\overset{->}{H} = {\left( \frac{j}{\omega \; \mu} \right){\nabla{\times \overset{->}{E}}}}},} & {{eq}\mspace{14mu} 10}\end{matrix}$

where ω is the angular frequency, and μ is the magnetic permeability.Usually, we assume μ=μ_(f).

In the grating region, eq. 10 gives

$\begin{matrix}{{\frac{\partial E_{gy}}{\partial z} = {j\; {\omega\mu}_{f}H_{gx}}},} & {{eq}.\mspace{14mu} 11} \\{{\frac{\partial H_{gx}}{\partial z} = {{j\; {\omega ɛ}_{f}{ɛ(x)}E_{gy}} + \frac{\partial H_{gz}}{\partial x}}},} & {{eq}.\mspace{14mu} 12}\end{matrix}$

Substituting eqs. 5 and 6 into eqs. 11 and 12 leads to

$\begin{matrix}{{\frac{\partial S_{yi}}{\partial z} = {k_{0}U_{xi}}},} & {{eq}.\mspace{14mu} 13} \\{{\frac{\partial U_{xi}}{\partial z} = {{\left( \frac{k_{xi}^{2}}{k_{0}} \right)S_{yi}} - {k_{0}{\sum\limits_{p = {- \infty}}^{\infty}{ɛ_{({i - p})}S_{yp}}}}}},} & {{eq}.\mspace{14mu} 14}\end{matrix}$

which are the set of coupled equations to be solved for the spatialharmonic components of the fields, S_(yi), and U_(xi).

When put in matrix form, eqs. 13 and 14 are

$\begin{matrix}{\begin{bmatrix}\frac{\partial S_{y}}{\partial\left( z^{\prime} \right)} \\\frac{\partial U_{x}}{\partial\left( z^{\prime} \right)}\end{bmatrix} = {\begin{bmatrix}0 & I \\A & 0\end{bmatrix}\begin{bmatrix}S_{y} \\U_{x}\end{bmatrix}}} & {{eq}.\mspace{14mu} 15}\end{matrix}$

where z′=k₀z . In eq. 15,

A=K _(x) ² −E,   eq. 16

K_(x) is a diagonal matrix with elements k_(xi)/k₀, and E is thepermittivity matrix, whose elements consist of the permittivity harmoniccomponents:

E _(i,j)=ε_((i-j)).   eq. 17

The permittivity matrix, E, should not be confused with the electricfield, which will always have a Cartesian component subscript.

Equation 15 can be further reduced to

$\begin{matrix}{\left\lbrack \frac{\partial^{2}S_{y}}{\partial\left( z^{\prime} \right)^{2}} \right\rbrack = {{\lbrack A\rbrack \left\lbrack S_{y} \right\rbrack}.}} & {{eq}.\mspace{14mu} 18}\end{matrix}$

Eq. 18 is in practice truncated after order N, which corresponds toretaining 2N+1 spatial harmonic terms in all of the Fourier series(positive and negative orders plus the zero term), leaving 2N+1 columnvectors for S_(y) and

$\left\lbrack \frac{\partial^{2}S_{y}}{\partial\left( z^{\prime} \right)^{2}} \right\rbrack,$

and a (2N+1)×(2N+1) matrix A.

The general solution for eq. 18, for a given truncation order N, can beexpressed in terms of the eigenvalues and eigenvectors of the matrix A:

$\begin{matrix}{{S_{yi} = {\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\{ {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}z} \right)}} + {c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},} & {{eq}.\mspace{14mu} 19} \\{{U_{xi} = {\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\{ {{{- c_{m}^{+}}{\exp \left( {{- k_{0}}q_{m}z} \right)}} + {c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},{{{where}\mspace{14mu} V} = {{WQ}.}}} & {{eq}.\mspace{14mu} 20}\end{matrix}$

Q is a diagonal matrix with elements q_(m), which are the square rootsof the 2N+1 eigenvalues of the matrix A, and W is the (2N+1)×(2N+1)matrix formed by the corresponding eigenvectors, with elements w_(i,m).

The coefficients c_(m) ⁺, and c_(m) ⁻ are determined, along with thereflected and diffracted field amplitudes, by matching the tangentialelectric and magnetic fields at the boundaries between the two regions,z=0 and z=d (see FIG. 1).

At the z=0 boundary, eqs. 4 and 9 imply that

$\begin{matrix}\begin{matrix}{{E_{I,y}_{z = 0}} = {{\exp \left\lbrack {{- j}\; k_{x\; 0}x} \right\rbrack} + {\sum\limits_{i = {- N}}^{N}{R_{i}{\exp \left( {{- j}\; k_{xi}x} \right)}}}}} \\{= {\sum\limits_{i = {- N}}^{N}{{S_{yi}(0)}{\exp \left( {{- j}\; k_{xi}x} \right)}}}} \\{= {\sum\limits_{i = {- N}}^{N}{{\exp \left( {{- j}\; k_{xi}x} \right)}{\left\{ {\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\lbrack {c_{m}^{+} + {c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}} \right\rbrack}} \right\}.}}}}\end{matrix} & {{eq}.\mspace{14mu} 21}\end{matrix}$

For the equality to hold in eq. 21, each of the components must beequal, so that

$\begin{matrix}{{\delta_{i\; 0} + R_{i}} = {\sum\limits_{m = 1}^{{2N} + 1}{{w_{i,m}\left\lbrack {c_{m}^{+} + {c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}} \right\rbrack}.}}} & {{eq}.\mspace{14mu} 22}\end{matrix}$

A similar argument can be applied to the magnetic field, which leads to

$\begin{matrix}{{{j\left\lbrack {{n_{I}\cos \; {\theta\delta}_{i\; 0}} - {\left( \frac{k_{I,{zi}}}{k_{0}} \right)R_{i}}} \right\rbrack} = {\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\lbrack {c_{m}^{+} - {c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}} \right\rbrack}}},} & {{eq}.\mspace{14mu} 23}\end{matrix}$

at the z=0 boundary, where the magnetic field in region I was obtainedfrom

$\begin{matrix}{{H_{I,x}_{z = 0}} = {{{{- \left( \frac{j}{\omega\mu} \right)}\frac{\partial E_{I,y}}{\partial z}}_{z = 0}} = {\left( \frac{j}{\omega\mu} \right){\left\{ {{j\; k_{0}n_{I}\cos \; {{\theta exp}\left\lbrack {{- j}\; k_{x\; 0}x} \right\rbrack}} - {\sum\limits_{i = {- N}}^{N}{j\; k_{I,{zi}}R_{i}{\exp \left( {{- j}\; k_{xi}x} \right)}}}} \right\}.}}}} & {{eq}.\mspace{14mu} 24}\end{matrix}$

Note that it is also necessary to use the relationship

${{\frac{1}{\omega \; \mu_{0}}\sqrt{\frac{\mu_{0}}{ɛ_{0}}}} = {\frac{c}{\omega} = \frac{1}{k_{0}}}},$

where c is the speed of light in vacuum, when deriving eq. 23.

Eqs. 22 and 23 can be put in matrix form:

$\begin{matrix}{{{\begin{bmatrix}\delta_{i\; 0} \\{j\; n_{1}\cos \; {\theta\delta}_{i\; 0}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\; Y_{I}}\end{bmatrix}\lbrack R\rbrack}} = {\begin{bmatrix}W & {WX} \\V & {- {VX}}\end{bmatrix}\begin{bmatrix}c^{+} \\c^{-}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 25}\end{matrix}$

where Y₁ and X are diagonal matrices with elements (k_(I,zi)/k₀) andexp(−k₀q_(m)d), respectively.

At the z=d boundary,

$\begin{matrix}{{{\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\lbrack {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} + c_{m}^{-}} \right\rbrack}} = T_{i}},} & {{eq}.\mspace{14mu} 26} \\{{{\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\lbrack {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} + c_{m}^{-}} \right\rbrack}} = {{j\left( {k_{{II},{zi}}/k_{0}} \right)}T_{i}}},} & {{eq}.\mspace{14mu} 27} \\{{{\begin{bmatrix}{WX} & W \\{VX} & {- V}\end{bmatrix}\begin{bmatrix}c^{+} \\c^{-}\end{bmatrix}} = {\begin{bmatrix}I \\{j\; Y_{II}}\end{bmatrix}\lbrack T\rbrack}},} & {{eq}.\mspace{14mu} 28}\end{matrix}$

where Y_(II) is a diagonal matrix with elements (k_(IIzi)/k₀).

Equations 25 and 28 are solved simultaneously for the coefficients c_(m)⁺, and c_(m) ⁻, and diffracted amplitudes R, and T_(i). It should bepointed out that there are many ways to solve the boundary equations.Here we will outline an efficient implementation of the enhancedtransmission matrix—partial solution approach from M. G. Moharam, D. A.Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of therigorous coupled-wave analysis for surface-relief gratings: enhancedtransmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077-1086 (1995),which can be used if only the reflected amplitudes are desired.Rewriting eq. 28,

$\begin{matrix}{{{\begin{bmatrix}{WX} \\{VX}\end{bmatrix}c^{+}} = {\begin{bmatrix}{- W} & I \\V & {j\; Y_{II}}\end{bmatrix}\begin{bmatrix}c^{-} \\T\end{bmatrix}}},} & {{eq}.\mspace{14mu} 29} \\{{{\begin{bmatrix}{- W} & I \\V & {j\; Y_{II}}\end{bmatrix}^{- 1}\begin{bmatrix}{WX} \\{VX}\end{bmatrix}}c^{+}} = {\begin{bmatrix}c^{-} \\T\end{bmatrix}.}} & {{eq}.\mspace{14mu} 30}\end{matrix}$

The top half of the matrix on the left side of eq. 30 is redefined as anew matrix, a:

$\begin{matrix}{\begin{bmatrix}a \\b\end{bmatrix} \equiv {\begin{bmatrix}{- W} & I \\V & {j\; Y_{II}}\end{bmatrix}^{- 1}\begin{bmatrix}{WX} \\{VX}\end{bmatrix}}} & {{eq}.\mspace{14mu} 31}\end{matrix}$

so that

ac⁺=c⁻.   eq. 32

This allows us to rewrite eq. 25 as

$\begin{matrix}{{{\begin{bmatrix}\delta_{i\; 0} \\{j\; n_{I}\cos \; {\theta\delta}_{i\; 0}}\end{bmatrix} + {\begin{bmatrix}I \\{j\; Y_{I}}\end{bmatrix}\lbrack R\rbrack}} = {{\begin{bmatrix}{W\left( {I + {Xa}} \right)} \\{V\left( {I - {Xa}} \right)}\end{bmatrix}c^{+}} = {\begin{bmatrix}f \\g\end{bmatrix}c^{+}}}},{{{where}\mspace{14mu} f} \equiv {{W\left( {I + {Xa}} \right)}\mspace{14mu} {and}\mspace{14mu} g} \equiv {{V\left( {I - {Xa}} \right)}.}}} & {{eq}.\mspace{14mu} 33}\end{matrix}$

We first solve eq. 33 for the c⁺ by multiplying the top half of theequations by jY_(I) and adding that to the bottom half to eliminate R:

[jY _(I) f+g]c ⁺ =j(Y _(I))_(0,0)δ_(i0) +jn _(I) cos θδ_(i0),   eq. 34

which is a (2N+1)×(2N+1) system of equations that are solved for thec_(m) ⁺. Note that (Y_(I))_(0,0) refers to the center element of thematrix Y_(I), (k_(I,z0)/k₀). The reflected amplitudes are then given by

R=fc ⁺−δ_(i0).   eq. 35

The diffracted efficiencies are obtained from

$\begin{matrix}{{DE}_{ri} = {R_{i}R_{i}^{*}\; {{{Re}\left( \frac{k_{I,{zi}}}{k_{0}n_{I}\cos \; \theta} \right)}.}}} & {{eq}.\mspace{14mu} 36}\end{matrix}$

Note that for the zero order at normal incidence,

R _(TE) ≡DE _(r0) =R ₀R₀*.   eq. 37

The partial solution approach can be generalized to L layers if we startwith f_(L+1)=I and g_(L+1)=jY_(II), where L+1 refers to the substrate,and use

$\begin{matrix}{{\begin{bmatrix}a_{L} \\b_{L}\end{bmatrix} \equiv {\begin{bmatrix}{- W_{L}} & f_{L + 1} \\V_{L} & g_{L + 1}\end{bmatrix}^{- 1}\begin{bmatrix}{W_{L}X_{L}} \\{V_{L}X_{L}}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 38}\end{matrix}$

where W_(L), V_(L), come from the eigen-problem for layer L, but areotherwise defined as in the single layer case, andX_(L)=exp(−k₀q_(m,L)d_(L)), with d_(L) defined as the layer thickness.

Then we define

$\begin{matrix}{{\begin{bmatrix}f_{L} \\g_{L}\end{bmatrix} \equiv \begin{bmatrix}{W_{L}\left( {I + {X_{L}a_{L}}} \right)} \\{V_{L}\left( {I - {X_{L}a_{L}}} \right)}\end{bmatrix}},} & {{eq}.\mspace{14mu} 39}\end{matrix}$

substitute eq. 39 back into eq. 38 for the L−1 layer, and repeat theprocess until we obtain f₁ and g₁:

$\begin{matrix}\begin{matrix}{{\begin{bmatrix}\delta_{i\; 0} \\{j\; n_{I}\cos \; {\theta\delta}_{i\; 0}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\; Y_{I}}\end{bmatrix}\lbrack R\rbrack}} = {\begin{bmatrix}{W_{1}\left( {I + {X_{1}a_{1}}} \right)} \\{V_{1}\left( {I - {X_{1}a_{1}}} \right)}\end{bmatrix}c_{1}^{+}}} \\{{= {\begin{bmatrix}f_{1} \\g_{1}\end{bmatrix}c_{1}^{+}}},}\end{matrix} & {{eq}.\mspace{14mu} 40}\end{matrix}$

which gives

[jY _(I) f ₁ +g ₁ ]c ₁ ⁺ =j(Y _(I))_(0,0)δ_(i0) +jn _(I) cos θδ_(i0).  eq. 41

Eq. 41 is solved for c₁ ⁺, and the reflectance amplitudes are then

R=f ₁ c ₁ ⁺−δ_(i0).   eq. 42

For the TM incident case, the magnetic field has only a y-component,while the electric field has x- and z-components. In the incidentmedium,

$\begin{matrix}{{H_{{inc},y} = {\exp \left\lbrack {{- j}\; k_{0}{n_{I}\left( {{\sin \; \theta \; x} + {\cos \; \theta \; z}} \right)}} \right\rbrack}},} & {{eq}.\mspace{14mu} 43} \\{H_{I,y} = {H_{{inc},y} + {\sum\limits_{i = {- \infty}}^{\infty}{R_{i}{{\exp \left\lbrack {- {j\left( {k_{xi} - {k_{I,{zi}}z}} \right)}} \right\rbrack}.}}}}} & {{eq}.\mspace{14mu} 44}\end{matrix}$

In the substrate medium,

$\begin{matrix}{H_{{II},y} = {\sum\limits_{i = {- \infty}}^{\infty}{T_{i}\exp {\left\{ {- {j\left\lbrack {{k_{xi}x} + {k_{{II},{zi}}\left( {z - d} \right)}} \right\rbrack}} \right\}.}}}} & {{eq}.\mspace{14mu} 45}\end{matrix}$

The tangential fields in the grating region are

$\begin{matrix}{H_{gy} = {\sum\limits_{i = {- \infty}}^{\infty}{{U_{yi}(z)}{\exp \left( {{- j}\; k_{xi}x} \right)}}}} & {{eq}.\mspace{14mu} 46} \\{E_{gx} = {{j\left( \frac{\mu_{0}}{ɛ_{0}} \right)}^{1/2}{\sum\limits_{i = {- \infty}}^{\infty}{{S_{xi}(z)}{{\exp \left( {{- j}\; k_{xi}x} \right)}.}}}}} & {{eq}.\mspace{14mu} 47}\end{matrix}$

The fields satisfy Maxwell's equation:

$\begin{matrix}{{\overset{\rightarrow}{E} = {\left( \frac{- j}{{\omega ɛ}_{f}n^{2}} \right){\nabla{\times \overset{\rightarrow}{H}}}}},} & {{eq}.\mspace{14mu} 48}\end{matrix}$

which leads to

$\begin{matrix}{{\frac{\partial H_{gy}}{\partial z} = {{- j}\; {\omega ɛ}_{f}{ɛ(x)}E_{gx}}},} & {{eq}.\mspace{14mu} 49} \\{\frac{\partial E_{gx}}{\overset{\Cap}{o}z} = {{{- {j\omega\mu}_{f}}H_{gy}} + \frac{\overset{\Cap}{o}E_{gx}}{\partial x}}} & {{eq}.\mspace{14mu} 50}\end{matrix}$

in the grating region. Eqs. 49 and 50 can be written in matrix form:

$\begin{matrix}{{\begin{bmatrix}{{\partial U_{y}}/{\partial\left( z^{\prime} \right)}} \\{{\partial S_{x}}/{\partial\left( z^{\prime} \right)}}\end{bmatrix} = {\begin{bmatrix}0 & E \\B & 0\end{bmatrix}\begin{bmatrix}U_{y} \\S_{x}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 51}\end{matrix}$

where z′=k₀z , and

B=K _(x) E ⁻¹ K _(x) −I.   eq. 52

K_(x) and E are defined as before. Here we add a modification proposedindependently by Lalanne and Morris (P. Lalanne and G. M. Morris,“Highly improved convergence of the coupled-wave method for TMpolarization,” J. Opt. Soc. Am. A 13, 779-784 (1996)), and Granet andGuizal (G. Granet and B. Guizal, “Efficient implementation of thecoupled-wave method for metallic lamellar gratings in TM polarization,”J. Opt. Soc. Am. A 13, 1019-1023 (1996)):

$\begin{matrix}{{\begin{bmatrix}{{\partial U_{y}}/{\partial\left( z^{\prime} \right)}} \\{{\partial S_{x}}/{\partial\left( z^{\prime} \right)}}\end{bmatrix} = {\begin{bmatrix}0 & {Einv}^{- 1} \\B & 0\end{bmatrix}\begin{bmatrix}U_{y} \\S_{x}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 53}\end{matrix}$

where Einv⁻¹ is the inverse of the inverse permittivity matrix, Einv,with (Einv)_(ij)=(1/ε)_(ij)=α_((i-j)), where α_(m) are the Fouriercoefficients of the inverse of the permittivity function. Themodification of eq. 53 improves the convergence rate for the TM casesignificantly, especially for metallic materials.

Eq. 53 can be reduced to

└∂² U _(y)/∂(z′)² ┘=└Einv⁻¹ B[]U _(y)┘.   eq. 54

Eq. 54 is solved in terms of the eigenvalues and eigenvectors of thematrix Einv⁻¹B, which gives for truncation order N

$\begin{matrix}{{{U_{yi}(z)} = {\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\{ {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}z} \right)}} + {c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},} & {{eq}.\mspace{14mu} 55} \\{{{S_{xi}(z)} = {\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\{ {{{- c_{m}^{+}}{\exp \left( {{- k_{0}}q_{m}z} \right)}} + {c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}} \right\}}}},{{{where}\mspace{14mu} V} = {{Einv}\; {{WQ}.}}}} & {{eq}.\mspace{14mu} 56}\end{matrix}$

Again, Q is a diagonal matrix with elements q_(m), which are the squareroots of the 2N+1 eigenvalues of the matrix Einv⁻¹B, and W is the(2N+1)×(2N+1) matrix formed by the corresponding eigenvectors, withelements w_(i,m).

The tangential fields are matched at the two boundaries in a similarmanner as before, leading to

$\begin{matrix}{{{\delta_{i\; 0} + R_{i}} = {\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\lbrack {c_{m}^{+} + {c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}} \right\rbrack}}},} & {{eq}.\mspace{14mu} 57} \\{{{j\left\lbrack {{\left( \frac{\cos \; \theta}{n_{I}} \right)\delta_{i\; 0}} - {\left( \frac{k_{I,{zi}}}{k_{0}n_{I}^{2}} \right)R_{i}}} \right\rbrack} = {\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\lbrack {c_{m}^{+} - {c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}} \right\rbrack}}},} & {{eq}.\mspace{14mu} 58}\end{matrix}$

or in matrix form:

$\begin{matrix}{{{\begin{bmatrix}\delta_{i\; 0} \\{j\; \delta_{i\; 0}\cos \; {\theta/n_{I}}}\end{bmatrix} + {\begin{bmatrix}I \\{{- j}\; Z_{I}}\end{bmatrix}\lbrack R\rbrack}} = {\begin{bmatrix}W & {WX} \\V & {- {VX}}\end{bmatrix}\begin{bmatrix}c^{+} \\c^{-}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 59}\end{matrix}$

where Z_(I) and X are diagonal matrices with elements (k_(I,zi)/k₀n_(I)²) and exp(−k₀q_(m)d), respectively.

At the z=d boundary,

$\begin{matrix}{{{\sum\limits_{m = 1}^{{2N} + 1}{w_{i,m}\left\lbrack {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} + c_{m}^{-}} \right\rbrack}} + T_{i}},} & {{eq}.\mspace{14mu} 60} \\{{{\sum\limits_{m = 1}^{{2N} + 1}{v_{i,m}\left\lbrack {{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} + c_{m}^{-}} \right\rbrack}} = {{j\left( \frac{k_{{II},{zi}}}{k_{0}n_{II}^{2}} \right)}T_{i}}},} & {{eq}.\mspace{14mu} 61}\end{matrix}$

or in matrix form:

$\begin{matrix}{{{\begin{bmatrix}{WX} & W \\{VX} & {- V}\end{bmatrix}\begin{bmatrix}c^{+} \\c^{-}\end{bmatrix}} = {\begin{bmatrix}I \\{j\; Z_{II}}\end{bmatrix}\lbrack T\rbrack}},} & {{eq}.\mspace{14mu} 62}\end{matrix}$

where Z_(II) is a diagonal matrix with elements (k_(II,zi)/k₀n_(II) ²).

The boundary problem is solved in the same manner as before, giving

$\begin{matrix}{{{\left\lbrack {{j\; Z_{I}f} + g} \right\rbrack c^{+}} = {{{j\left( Z_{I} \right)}_{0,0}\delta_{i\; 0}} + {j\frac{\cos \; \theta}{n_{I}}\delta_{i\; 0}}}},} & {{eq}.\mspace{14mu} 63}\end{matrix}$

for the coefficients c⁺, and finally

R=fc ⁺−δ_(i0)   eq. 64

for the reflected amplitudes.

For L layers, the recursion is the same as in the TE case, giving

$\begin{matrix}{{\left\lbrack {{j\; Z_{I}f_{1}} + g_{1}} \right\rbrack c_{1}^{+}} = {{{j\left( Z_{I} \right)}_{0,0}\delta_{i\; 0}} + {j\frac{\cos \; \theta}{n_{I}}{\delta_{i\; 0}.}}}} & {{eq}.\mspace{14mu} 65}\end{matrix}$

for c₁ ⁺, and

R=f ₁ c ₁ ⁺−δ_(i0).   eq. 66

for the diffracted amplitudes.

The preceding description can be applied to polarized reflectance datacollected in the phi=0 mount, or to un-polarized reflectance by use ofeq. 1. Typically, the reflectance data is used to optimize theparameters of a theoretical model representative of the presumedstructure, using one of many common algorithms, such as theLevenberg-Marquardt or Simplex algorithms (see W. H. Press, S. A.Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C(2^(nd) Edition), Cambridge University Press, Cambridge, 1992, forexample). The model calculation is performed at each regression stepusing the above RCW formulation. Alternately, a library database ofspectra corresponding to the entire parameter space expected for themodel is pre-generated. In this case the regression retrieves therequired spectrum from the library at each step, or any of a variety ofsearch mechanisms are used during the measurement to find the best matchto the actual reflectance.

The RCW calculation is dominated by a (2N+1)×(2N+1) eigen-problem and a(2N+1)×(2N+1) boundary problem, as well as several (2N+1)×(2N+1) matrixmultiplications. All of these operations are order n³, where n is thematrix size, which means that doubling the truncation order results inan approximately 8-fold increase in overall computation time. For largetruncation order the calculation time can become significant.

The truncation order is dependent on the structure being simulated.Generally, larger pitch to wavelength ratios and larger contrast betweenline and space optical properties will require larger truncation orderto converge. In addition, complicated feature profiles can require alarge number of layer slices to correctly approximate the line shape.All said, for some structures the required RCW calculations can becomeprohibitively time consuming.

A reduction of the required computation steps for a given truncationorder, N, will directly address the issue, since it will reduce thecalculation time everywhere. In particular, the order n³ behavior of theRCW method means that reducing the matrix sizes for given truncationorder can have a dramatic effect on the computation speed. It ispossible to do this by exploiting the symmetry for certain incidence andgrating conditions.

A method for reducing the matrix size from 2N+1 to N+1 in the case ofnormal incidence will now be described.

For normal incidence, we have

k _(xi) =−ik ₀(λ₀/Λ)=−2πi/Λ,   eq. 67

k_(xi)=−k_(x-i)   eq. 68

k_(I,zi)=k_(I,z-i)   eq. 69

from eqs. 7 and 8. If the grating is symmetric, there are additionalconditions imposed on the fields:

E_(y,i)=E_(y,−i)   eq. 70

H_(x,i)=H_(x,−i)   eq. 71

for TE polarization, and

E_(x,i)=E_(x,−i)   eq. 72

H_(y,i)=H_(y,−i.)   eq. 73

for TM polarization, where the subscript i refers to the expansion term,which in the incident region, corresponds to the diffraction order.

These conditions can be reasoned out from symmetry arguments, or byexperimentation with calculations performed using the more general RCWmethod presented above. The symmetry relationships are valid in allregions of the grating problem. There is also a 180 degree phasedifference between opposite odd orders, but this can be ignored when notconsidering interference between multiple gratings.

Applying the above relations to the various regions in the 2-D gratingproblem for the TE case:

R_(,i)=R_(i)   eq. 74

T_(,i)=T_(−i)   eq. 75

in regions I and II, and

S_(yi)=S_(y,−i)   eq. 76

U_(xi)=U_(x,−i)   eq. 77

for the fields in the grating region. These conditions can be applieddirectly to the Fourier expansions of eqs. 2-6.

We first note that eq. 2, the incident wave has no x-dependence, and issimply

E _(inc,y)=exp(−jk ₀n_(I)z).   eq. 78

Applying eqs. 74 and 78 to eq. 3 gives

$\begin{matrix}\begin{matrix}{E_{I,y} = {E_{{inc},y} +}} \\{{\sum\limits_{i = {- \infty}}^{\infty}\; {R_{i}{\exp \left\lbrack {- {j\left( {{k_{xi}x} - {k_{I,{zi}}z}} \right)}} \right\rbrack}}}} \\{= {E_{{inc},y} + {R_{0}{\exp \left( {j\; k_{I,{z\; 0}}z} \right)}} +}} \\{{{\sum\limits_{i = 1}^{\infty}\; {R_{i}{\exp \left\lbrack {- {j\left( {{k_{xi}x} - {k_{I,{zi}}z}} \right)}} \right\rbrack}}} +}} \\{{\sum\limits_{i = {- \infty}}^{- 1}\; {R_{i}{\exp \left\lbrack {- {j\left( {{k_{xi}x} - {k_{I,{zi}}z}} \right)}} \right\rbrack}}}} \\{= {E_{{inc},y} + {R_{0}{\exp \left( {j\; k_{I,{z\; 0}}z} \right)}} +}} \\{{{\sum\limits_{i = 1}^{\infty}\; {R_{i}{\exp \left\lbrack {- {j\left( {{k_{xi}x} - {k_{I,{zi}}z}} \right)}} \right\rbrack}}} +}} \\{{\sum\limits_{i = 1}^{\infty}\; {R_{- i}{\exp \left\lbrack {- {j\left( {{k_{x,{- i}}x} - {k_{I,z,{- i}}z}} \right)}} \right\rbrack}}}} \\{= {E_{{inc},y} + {R_{0}{\exp \left( {j\; k_{I,{z\; 0}}z} \right)}} +}} \\{{{\sum\limits_{i = 1}^{\infty}\; {R_{i}{\exp \left\lbrack {{{- j}\; k_{xi}x} + {j\; k_{I,{zi}}z}} \right\rbrack}}} +}} \\\left. {\sum\limits_{i = 0}^{\infty}\; {R_{i}{\exp\left\lbrack {{j\; k_{xi}x} + {j\; k_{I,{zi}}z}} \right)}}} \right\rbrack \\{= {E_{{inc},y} + {R_{0}{\exp \left( {j\; k_{I,{z\; 0}}z} \right)}} +}} \\{{\sum\limits_{i = 1}^{\infty}\; {R_{i}{{\exp \left( {j\; k_{I,{zi}}z} \right)}\left\lbrack {{\exp \left( {{- j}\; k_{xi}x} \right)} + {\exp \left( {j\; k_{xi}x} \right)}} \right\rbrack}}}} \\{= {E_{{inc},y} + {R_{0}{\exp \left( {j\; k_{I,{z\; 0}}z} \right)}} +}} \\{{\sum\limits_{i = 1}^{\infty}\; {2R_{i}{\exp \left( {j\; k_{I,{zi}}z} \right)}{{\cos \left( {j\; k_{xi}x} \right)}.}}}}\end{matrix} & {{eq}.\mspace{14mu} 79}\end{matrix}$

At z=0, eq. 79 becomes simply a Fourier cosine series expansion of thefield at the boundary:

$\begin{matrix}{{E_{I,y}\left( {z = 0} \right)} = {{1 + R_{0} + {\sum\limits_{i = 1}^{\infty}\; {2R_{i}{\cos \left( {j\; k_{xi}x} \right)}}}} = {1 + R_{0} + {\sum\limits_{i = 1}^{\infty}\; {2R_{i}{{\cos \left( {j\; 2\pi \; {{ix}/\Lambda}} \right)}.}}}}}} & {{eq}.\mspace{14mu} 80}\end{matrix}$

Similarly, at the z=d boundary:

$\begin{matrix}{{{E_{{II},y}\left( {z = d} \right)} = {T_{0} + {\sum\limits_{i = 1}^{\infty}\; {2T_{i}{\cos \left( {j\; k_{xi}x} \right)}}}}},} & {{eq}.\mspace{14mu} 81}\end{matrix}$

And inside the grating region,

$\begin{matrix}{{E_{gy} = {{S_{y\; 0}(z)} + {\sum\limits_{i = 1}^{\infty}\; {2{S_{yi}(z)}{\cos \left( {j\; k_{xi}x} \right)}}}}},} & {{eq}.\mspace{14mu} 82} \\{H_{gx} = {{- {j\left( \frac{ɛ_{f}}{\mu_{f}} \right)}^{1/2}}{\left\{ {{U_{x\; 0}(z)} + {\sum\limits_{i = 1}^{\infty}\; {2{U_{xi}(z)}{\cos \left( {j\; k_{xi}x} \right)}}}} \right\}.}}} & {{eq}.\mspace{14mu} 83}\end{matrix}$

The fact that the fields can be reduced to cosine series is a directconsequence of the even symmetry of the diffraction problem with respectto the x coordinate under the normal incidence condition. For a giventruncation order, N, the reduced expansions contain exactly the sameinformation, but with N+1 unknowns instead of 2N+1.

To show that the form of the boundary problem need not be modified, wefirst assume that the size of the eigen-problem can be reduced to(N+1)×(N+1), and otherwise has the same form as eqs. 19 and 20, as shownbelow. Matching the y-components of the electric field at the z=0boundary:

$\begin{matrix}\begin{matrix}{{1 + R_{0} + {\sum\limits_{i = 1}^{\infty}\; {2R_{i}{\cos \left( {j\; k_{xi}x} \right)}}}} = {{S_{y\; 0}(0)} + {\sum\limits_{i = 1}^{\infty}\; {2{S_{yi}(0)}{\cos \left( {j\; k_{xi}x} \right)}}}}} \\{= {{\sum\limits_{m = 1}^{N + 1}\; {w_{0,m}\begin{bmatrix}{c_{m}^{+} +} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}} +}} \\{{\sum\limits_{i = 1}^{N}\; {2{\cos \left( {j\; k_{xi}x} \right)}}}} \\{{\sum\limits_{m = 1}^{N + 1}\; {{w_{i,m}\begin{bmatrix}{c_{m}^{+} +} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}.}}}\end{matrix} & {{eq}.\mspace{14mu} 84}\end{matrix}$

Since this condition holds for all x, terms with the same cos(jk_(xi)x)on each side must be equal:

$\begin{matrix}{{1 + R_{0}} = {\sum\limits_{m = 1}^{N + 1}\; {w_{0,m}\begin{bmatrix}{c_{m}^{+} +} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}}} & {{eq}.\mspace{14mu} 85} \\{{{2R_{i}} = {\sum\limits_{i = 1}^{N + 1}\; {2{w_{i,m}\begin{bmatrix}{c_{m}^{+} +} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}}}},} & {{eq}.\mspace{14mu} 86} \\{or} & \; \\{{\delta_{i\; 0} + R_{i}} = {\sum\limits_{m = 1}^{N + 1}\; {w_{i,m}\begin{bmatrix}{c_{m}^{+} +} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}}} & {{eq}.\mspace{14mu} 87}\end{matrix}$

Which is the same as eq. 22, except that the index i runs from 0 to N,and the index m runs from 1 to N+1.

The magnetic field at the z=0 boundary in region I is

$\begin{matrix}\begin{matrix}{{H_{I,x}_{z = 0}} = {{{- \left( \frac{j}{\omega\mu} \right)}\frac{\partial E_{I,y}}{\partial z}}_{z = 0}}} \\{{= {\left( \frac{j}{\omega\mu} \right)\begin{bmatrix}{{j\; k_{0}n_{I}} -} \\{{j\; k_{0}n_{I}R_{0}} -} \\{\sum\limits_{i = 1}^{\infty}{j\; k_{I,{zi}}2R_{i}{\cos \left( {j\; k_{xi}x} \right)}}}\end{bmatrix}}},}\end{matrix} & {{eq}.\mspace{14mu} 88}\end{matrix}$

which leads to

$\begin{matrix}{{j\left\lbrack {{n_{I}\delta_{i\; 0}} - {\left( \frac{k_{I,{zi}}}{k_{0}} \right)R_{i}}} \right\rbrack} = {\sum\limits_{m = 1}^{N + 1}\; {v_{i,m}\begin{bmatrix}{c_{m}^{+} -} \\{c_{m}^{-}{\exp \left( {{- k_{0}}q_{m}d} \right)}}\end{bmatrix}}}} & {{eq}.\mspace{14mu} 89}\end{matrix}$

for the magnetic field condition.

Similarly, for the z=d boundary,

$\begin{matrix}{{{\sum\limits_{m = 1}^{N + 1}\; {w_{i,m}\begin{bmatrix}{{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} +} \\c_{m}^{-}\end{bmatrix}}} = T_{i}},} & {{eq}.\mspace{14mu} 90} \\{{\sum\limits_{m = 1}^{N + 1}\; {v_{i,m}\begin{bmatrix}{{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}d} \right)}} +} \\c_{m}^{-}\end{bmatrix}}} = {{j\left( \frac{k_{{II},{zi}}}{k_{0}n_{II}^{2}} \right)}{T_{i}.}}} & {{eq}.\mspace{14mu} 91}\end{matrix}$

Eqs. 87, 89, 90, and 91 obviously lead to the same boundary problem aseqs. 25 and 28, but now with (N+1)×(N+1) sets of equations instead of(2N+1)×(2N+1). The steps outlined in eqs. 29-42 can still be used tosolve the boundary problem. Alternately, since the form of the boundaryproblem is unchanged from the conventional formulation, any number ofother well-known techniques, such as the R-matrix, T-matrix, S-matrix,or the more recent enhanced R-matrix (E. L. Tan, “Enhanced R-matrixalgorithms for multilayered diffraction gratings,” Appl. Opt. 45,4803-4809 (2006) and hybrid-matrix algorithms (E. L. Tan, “Hybrid-matrixalgorithm for rigorous coupled-wave analysis of multilayered diffractiongratings,” J. Mod. Opt. 53, 417-428 (2006)), can be easily applied tothe reduced multiple layer diffraction problem. The form of the boundaryproblem differs from the reduction discussed in U.S. Pat. No. 6,898,537,in that the U.S. Pat. No. 6,898,537 teaches that every nonzerodiffracted reflectance coefficient must be multiplied by a factor oftwo.

To show how to reduce the eigen-problem to the form we assumed above, westart by applying eq. 76 to eq. 18. The rows of eq. 18 can be written inthe form

$\begin{matrix}{\frac{\partial^{2}S_{yi}}{\partial\left( z^{\prime} \right)^{2}} = {{\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{yi}} - {\sum\limits_{m = {- \infty}}^{\infty}\; {E_{i,m}{S_{ym}.}}}}} & {{eq}.\mspace{14mu} 92}\end{matrix}$

For the i=0 term,

$\begin{matrix}\begin{matrix}{\frac{\partial^{2}S_{y\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {- {\sum\limits_{m = {- \infty}}^{\infty}\; {E_{0,m}S_{ym}}}}} \\{= {{{- E_{0,0}}S_{y\; 0}} - {\sum\limits_{m = {- \infty}}^{- 1}\; {E_{0,m}S_{ym}}} - {\sum\limits_{m = 1}^{\infty}{E_{0,m}S_{ym}}}}} \\{= {{{- E_{0,0}}S_{y\; 0}} - {\sum\limits_{m = 1}^{\infty}\; {E_{0,{- m}}S_{ym}}} - {\sum\limits_{m = 1}^{\infty}{E_{0,m}S_{ym}}}}} \\{= {{{- E_{0,0}}S_{y\; 0}} - {\sum\limits_{m = 1}^{\infty}\; {\left( {E_{0,{- m}} + E_{0,m}} \right)S_{ym}}}}} \\{{= {{{- E_{0,0}}S_{y\; 0}} - {\sum\limits_{m = 1}^{\infty}\; {2\; E_{0,m}S_{ym}}}}},}\end{matrix} & {{eq}.\mspace{14mu} 92} \\{so} & \; \\{{\frac{\partial^{2}S_{y\; 0}}{\partial\left( z^{\prime} \right)^{2}} = {{{- E_{0,0}}S_{y\; 0}} - {\sum\limits_{m = 1}^{\infty}{2\; E_{0,m}S_{ym}}}}},{i = 0},} & {{eq}.\mspace{14mu} 93}\end{matrix}$

where we have used the fact that k_(x0)=0, and

E_(i,j)=E_(−i,−j),   eq. 94

which follows from eq. 17 for a symmetric grating.

For i♯0, we use eq. 76 to derive

$\begin{matrix}{{\frac{\partial^{2}S_{y\; i}}{\partial\left( z^{\prime} \right)^{2}} = \frac{\partial^{2}S_{y - i}}{\partial\left( z^{\prime} \right)^{2}}},} & {{eq}.\mspace{14mu} 95}\end{matrix}$

and add the i th and −i th rows:

$\begin{matrix}\begin{matrix}{{2\frac{\partial^{2}S_{y\; i}}{\partial\left( z^{\prime} \right)^{2}}} = {{\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} + {\left( \frac{k_{x - i}^{2}}{k_{0}^{2}} \right)S_{y - \; i}} - {\sum\limits_{m = {- \infty}}^{\infty}\; {E_{i,m}S_{ym}}} -}} \\{{\sum\limits_{m = {- \infty}}^{\infty}{E_{i,m}S_{ym}}}} \\{= {{\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} + {\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {\sum\limits_{m = {- \infty}}^{\infty}\; {E_{i,m}S_{ym}}} -}} \\{{\sum\limits_{m = {- \infty}}^{\infty}{E_{{- i},m}S_{ym}}}} \\{= {{2\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {E_{i,0}S_{y,0}} - {\sum\limits_{m = 1}^{\infty}{E_{i,m}S_{ym}}} -}} \\{{{\sum\limits_{m = 1}^{\infty}{E_{{- i},m}S_{ym}}} - {\sum\limits_{m = {- \infty}}^{- 1}{E_{{- i},m}S_{ym}}}}}\end{matrix} & {{eq}.\mspace{14mu} 96} \\{\mspace{79mu} \begin{matrix}{= {{2\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {2E_{i,0}S_{y,0}} - {\sum\limits_{m = 1}^{\infty}{E_{i,m}S_{ym}}} -}} \\{{{\sum\limits_{m = 1}^{\infty}{E_{i,{- m}}S_{ym}}} - {\sum\limits_{m = 1}^{\infty}{E_{{- i},m}S_{ym}}} - {\sum\limits_{m = 1}^{\infty}{E_{{- i},{- m}}S_{ym}}}}} \\{= {{2\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {2E_{i,0}S_{y,0}} - {2{\sum\limits_{m = 1}^{\infty}{E_{i,m}S_{ym}}}} -}} \\{{2{\sum\limits_{m = 1}^{\infty}{E_{i,{- m}}S_{ym}}}}} \\{{= {{2\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {2E_{i,0}S_{y,0}} - {2{\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{i,{- m}}} \right)S_{ym}}}}}},}\end{matrix}} & \; \\{giving} & \; \\{{\frac{\partial^{2}S_{y\; i}}{\partial\left( z^{\prime} \right)^{2}} = {{\left( \frac{k_{xi}^{2}}{k_{0}^{2}} \right)S_{y\; i}} - {E_{i,0}S_{y,0}} - {\sum\limits_{m = 1}^{\infty}{\left( {E_{i,m} + E_{i,{- m}}} \right)S_{ym}}}}},{i > 0.}} & \;\end{matrix}$

Eqs. 93 and 96 lead to matrix equations with the same form as eq. 18,but with (N+1)×(N+1) sized matrices instead of the original(2N+1)×(2N+1) sized matrices. The permittivity matrix E is replaced with

$\begin{matrix}{E_{reduced} = {\begin{bmatrix}E_{0,0} & {2E_{0,1}} & {2E_{0,2}} & \ldots \\E_{1,0} & \left( {E_{1,1} + E_{1,{- 1}}} \right) & \left( {E_{1,2} + E_{1,{- 2}}} \right) & \ldots \\E_{2,0} & \left( {E_{2,1} + E_{2,{- 1}}} \right) & \left( {E_{2,2} + E_{2,{- 2}}} \right) & \ldots \\\vdots & \; & \; & \ddots\end{bmatrix}.}} & {{eq}.\mspace{14mu} 97}\end{matrix}$

That is, the first column of the reduced matrix is replaced by E_(i,0)from the original matrix, and the other elements i,j areE_(i,j)+E_(i,−j), in terms of the elements of the old matrix, withi,j≧0. Eq. 97 can be compared with eq. 26 from the U.S. Pat. No.6,898,537, which does not include the E_(i,−j) term for reduced matrixelements when i+j is greater than the truncation order, N. This omissionis not suggested in that reference, since the corresponding unreducedmatrix includes the ε_(2N) permittivity coefficients for a given N.

The matrix K_(x) is simply replaced by an (N+1)×(N+1) diagonal matrixconsisting of the 0 and positive terms of the original K.

Therefore, we have for the new eigen-problem,

$\begin{matrix}{{\left\lbrack \frac{\partial^{2}S_{y\;}}{\partial\left( z^{\prime} \right)^{2}} \right\rbrack = {\left\lbrack A_{reduced} \right\rbrack \left\lbrack S_{y\;} \right\rbrack}},} & {{eq}.\mspace{14mu} 98} \\{with} & \; \\{A_{reduced} = {K_{x}^{2} - E_{reduced}}} & {{eq}.\mspace{14mu} 99}\end{matrix}$

Eq. 99 is solved in a manner similar to eq. 18:

$\begin{matrix}{S_{y\; i} = {\sum\limits_{m = 1}^{N + 1}{w_{i,m}\begin{Bmatrix}{{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}z} \right)}} +} \\{c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}\end{Bmatrix}}}} & {{eq}.\mspace{14mu} 100} \\{U_{xi} = {\sum\limits_{m = 1}^{N + 1}{v_{i,m}\begin{Bmatrix}{{{- c_{m}^{+}}{\exp \left( {{- k_{0}}q_{m}z} \right)}} +} \\{c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}\end{Bmatrix}}}} & {{eq}.\mspace{14mu} 101}\end{matrix}$

where V=WQ, Q is a diagonal matrix with elements q_(m), which are thesquare roots of the N+1 eigenvalues of the matrix A_(reduced), and W isthe (N+1)×(N+1) matrix formed by the corresponding eigenvectors, withelements

The size of the eigen-problem and boundary problem are therefore reducedto (N+1)×(N+1) for a given truncation order, N. Now we determine N+1reflected amplitudes, R_(i), i=0, . . . , N, but in light of the factthat R_(i)=R_(i), we have determined the same information as in theconventional formulation. It is important to note that no approximationswere made, except for the usual series truncations. For given truncationorder, N, the calculation gives an identical result, but is faster by afactor of approximately 8 compared to the standard RCW formulation.

For the TM case, just as for the TE case, the boundary problem reducesto the same form as the conventional formulation, but with an(N+1)×(N+1) system of equations instead of a (2N+1)×(2N+1) system.Again, we determine R_(i), i=0, . . . , N with a factor of 8 reductionin overall computation time. The task remaining is to determine thereduced eigen-problem for the TM case.

We again have eqs. 67-69, eqs. 74, 75, and 94, but this time use

S_(xi)=S_(x,−i)   eq. 102

U_(yi)=Y_(y,−i)   eq. 103

in the grating region. We can reduce the matrix Einv⁻¹B by applyingthese relations to eq. 54 directly, but this will lead to an unnecessary(2N+1)×(2N+1) matrix multiplication to find the elements of Einv⁻¹B. Weinstead go back to eq. 53 and reduce the matrices separately, resultingin an (N+1)×(N+1) multiplication instead.

We start with the first row of eq. 53, the rows of which are:

$\begin{matrix}{\frac{\partial U_{y\; i}}{\partial\left( z^{\prime} \right)} = {\sum\limits_{m = {- \infty}}^{\infty}{\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}}} & {{eq}.\mspace{14mu} 104} \\{{{{For}\mspace{14mu} i} = 0},\begin{matrix}{\frac{\partial U_{y\; 0}}{\partial\left( z^{\prime} \right)} = {{\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{0,m}S_{xm}}} + {\sum\limits_{m = {- \infty}}^{- 1}{\left( {Einv}^{- 1} \right)_{0,m}S_{xm}}}}} \\{= {{\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{0,m}S_{xm}}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{0,{- m}}S_{xm}}}}} \\{{= {{\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{\left( {Einv}^{- 1} \right)_{0,m} +} \\\left( {Einv}^{- 1} \right)_{0,{- m}}\end{bmatrix}S_{xm}}}}},}\end{matrix}} & \; \\{or} & \; \\{{\frac{\partial U_{y\; 0}}{\partial\left( z^{\prime} \right)} = {{\left( {Einv}^{- 1} \right)_{0,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{2\left( {Einv}^{- 1} \right)_{0,m}S_{xm}}}}},} & {{eq}.\mspace{14mu} 105} \\{{i = 0},} & \;\end{matrix}$

where we use

(Einv⁻¹)_(i,j)=(Einv⁻¹)_(−i,−j)   eq. 106

for a symmetric grating. For i♯0, we again add the i th and −i th rows:

$\begin{matrix}{{2\frac{\partial U_{y\; i}}{\partial\left( z^{\prime} \right)}} = {{\sum\limits_{m = {- \infty}}^{\infty}{\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}} + {\sum\limits_{m = {- \infty}}^{\infty}{\left( {Einv}^{- 1} \right)_{{- i},m}S_{xm}}}}} \\{= {{\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}} +}} \\{{{\sum\limits_{m = {- \infty}}^{- 1}{\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}} + {\left( {Einv}^{- 1} \right)_{{- i},0}S_{x\; 0}} +}} \\{{{\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{{- i},m}S_{xm}}} + {\sum\limits_{m = {- \infty}}^{- 1}{\left( {Einv}^{- 1} \right)_{{- i},m}S_{xm}}}}} \\{= {{\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\left( {Einv}^{- 1} \right)_{{- i},0}S_{x\; 0}} +}} \\{{{\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{i,{- m}}S_{xm}}} +}} \\{{{\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{{- i},m}S_{xm}}} + {\sum\limits_{m = 1}^{\infty}{\left( {Einv}^{- 1} \right)_{{- i},{- m}}S_{xm}}}}} \\{= {{2\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{2\left( {Einv}^{- 1} \right)_{i,m}S_{xm}}} +}} \\{{\sum\limits_{m = 1}^{\infty}{2\left( {Einv}^{- 1} \right)_{i,{- m}}S_{xm}}}} \\{= {{2\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{{2\begin{bmatrix}{\left( {Einv}^{- 1} \right)_{i,m} +} \\\left( {Einv}^{- 1} \right)_{i,{- m}}\end{bmatrix}}S_{xm}}}}}\end{matrix}$

so that

$\begin{matrix}{{\frac{\partial U_{y\; i}}{\partial\left( z^{\prime} \right)} = {{\left( {Einv}^{- 1} \right)_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{\left( {Einv}^{- 1} \right)_{i,m} +} \\\left( {Einv}^{- 1} \right)_{i,{- m}}\end{bmatrix}S_{xm}}}}},} & {{eq}.\mspace{14mu} 107} \\{i > 0.} & \;\end{matrix}$

In matrix form,

$\begin{matrix}{{Einv}_{reduced}^{- 1} = {\quad{\begin{bmatrix}\left( {Einv}^{- 1} \right)_{0,0} & {2\left( {Einv}^{- 1} \right)_{0,1}} & {2\left( {Einv}^{- 1} \right)_{0,2}} & \ldots \\\left( {Einv}^{- 1} \right)_{1,0} & \begin{bmatrix}{\left( {Einv}^{- 1} \right)_{1,1} +} \\\left( {Einv}^{- 1} \right)_{1,{- 1}}\end{bmatrix} & \begin{bmatrix}{\left( {Einv}^{- 1} \right)_{1,2} +} \\\left( {Einv}^{- 1} \right)_{1,{- 2}}\end{bmatrix} & \ldots \\\left( {Einv}^{- 1} \right)_{2,0} & \begin{bmatrix}{\left( {Einv}^{- 1} \right)_{2,1} +} \\\left( {Einv}^{- 1} \right)_{2,{- 1}}\end{bmatrix} & \begin{bmatrix}{\left( {Einv}^{- 1} \right)_{2,2} +} \\\left( {Einv}^{- 1} \right)_{2,{- 2}}\end{bmatrix} & \ldots \\\vdots & \; & \; & \ddots\end{bmatrix},}}} & {{eq}.\mspace{14mu} 108}\end{matrix}$

where the first column of the reduced matrix is replaced by(Einv⁻¹)_(0,0) from the original matrix, and the rest of the elementsi,j are (Einv⁻¹)_(ij)+(Einv⁻¹)_(i,−j), in terms of the elements of theold matrix, with i,j≧0.

For the matrix B, we reduce

$\begin{matrix}{\frac{\partial S_{x\; i}}{\partial\left( z^{\prime} \right)} = {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} - {U_{y\; i}.}}} & {{eq}.\mspace{14mu} 109} \\{{{{For}\mspace{14mu} i} = 0},} & \; \\{{\frac{\partial S_{x\; 0}}{\partial\left( z^{\prime} \right)} = {- U_{y\; 0}}},} & {{eq}.\mspace{14mu} 110}\end{matrix}$

since k_(x0)=0. For i♯0, adding the i th and −i th rows:

$\begin{matrix}\begin{matrix}{{2\frac{\partial S_{x\; i}}{\partial\left( z^{\prime} \right)}} = {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} +}} \\{{{\frac{k_{x - \; i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} - {2U_{y\; i}}}} \\{= {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}}} -}} \\{{{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = {- \infty}}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}} - {2U_{y\; i}}}}\end{matrix} & {{eq}.\mspace{14mu} 111} \\{\mspace{70mu} {= {{\frac{k_{x\; i}}{k_{0}}\begin{Bmatrix}{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} +} \\{{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} -} \\{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}} -} \\{\sum\limits_{m = {- \infty}}^{- 1}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}}\end{Bmatrix}} - {2U_{y\; i}}}}} & \; \\{\mspace{70mu} {= {{\frac{k_{x\; i}}{k_{0}}\begin{Bmatrix}{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} +} \\{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}} -} \\{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}} -} \\{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{x - m}}{k_{0}}U_{ym}}}\end{Bmatrix}} - {2U_{y\; i}}}}} & \; \\{\mspace{70mu} {= {{\frac{k_{x\; i}}{k_{0}}\begin{Bmatrix}{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,m}\frac{k_{xm}}{k_{0}}U_{ym}}} -} \\{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{i,{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}} -} \\{{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},m}\frac{k_{xm}}{k_{0}}U_{ym}}} +} \\{\sum\limits_{m = 1}^{\infty}{\left( E^{- 1} \right)_{{- i},{- m}}\frac{k_{xm}}{k_{0}}U_{ym}}}\end{Bmatrix}} - {2U_{y\; i}}}}} & \; \\\begin{matrix}{\mspace{70mu} {= {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{\left( E^{- 1} \right)_{i,m} +} \\{\left( E^{- 1} \right)_{{- i},{- m}} -} \\{\left( E^{- 1} \right)_{i,{- m}} -} \\\left( E^{- 1} \right)_{{- i},m}\end{bmatrix}\frac{k_{x\; m}}{k_{0}}U_{ym}}}} - {2U_{y\; i}}}}} \\{{= {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{{2\left( E^{- 1} \right)_{i,m}} -} \\{2\left( E^{- 1} \right)_{i,{- m}}}\end{bmatrix}\frac{k_{x\; m}}{k_{0}}U_{ym}}}} - {2U_{y\; i}}}},}\end{matrix} & \; \\{{or}{{\frac{\partial S_{x\; i}}{\partial\left( z^{\prime} \right)} = {{\frac{k_{x\; i}}{k_{0}}{\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{\left( E^{- 1} \right)_{i,m} -} \\\left( E^{- 1} \right)_{i,{- m}}\end{bmatrix}\frac{k_{x\; m}}{k_{0}}U_{ym}}}} - U_{y\; i}}},}} & \; \\{i > 0.} & \;\end{matrix}$

From eqs. 110 and 111, the reduced matrix for K_(x)E⁻¹K_(x) has zeros inthe first row and column, and the elements

${\frac{k_{x\; i}}{k_{0}}\begin{bmatrix}{\left( E^{- 1} \right)_{i,j} -} \\\left( E^{- 1} \right)_{i,{- j}}\end{bmatrix}}\frac{k_{x\; j}}{k_{0}}$

otherwise, with i,j>0. In matrix form,

$\begin{matrix}{\left( {K_{x}E^{- 1}K_{x}} \right)_{reduced} = {\quad\begin{bmatrix}0 & 0 & 0 & \ldots \\0 & {{\frac{k_{x\; 1}}{k_{0}}\begin{bmatrix}{\left( E^{- 1} \right)_{1,1} -} \\\left( E^{- 1} \right)_{1,{- 1}}\end{bmatrix}}\frac{k_{x\; 1}}{k_{0}}} & {{\frac{k_{x\; 1}}{k_{0}}\begin{bmatrix}{\left( E^{- 1} \right)_{1,2} -} \\\left( E^{- 1} \right)_{1,{- 2}}\end{bmatrix}}\frac{k_{x\; 2}}{k_{0}}} & \ldots \\0 & {{\frac{k_{x\; 2}}{k_{0}}\begin{bmatrix}{\left( E^{- 1} \right)_{2,1} -} \\\left( E^{- 1} \right)_{2,{- 1}}\end{bmatrix}}\frac{k_{x\; 1}}{k_{0}}} & {{\frac{k_{x\; 2}}{k_{0}}\begin{bmatrix}{\left( E^{- 1} \right)_{2,2} -} \\\left( E^{- 1} \right)_{2,{- 2}}\end{bmatrix}}\frac{k_{x\; 2}}{k_{0}}} & \ldots \\\vdots & \; & \; & \ddots\end{bmatrix}}} & {{eq}.\mspace{14mu} 112}\end{matrix}$

This gives

B _(reduced)=(K _(x) E ⁻¹ K _(x))_(reduced) −I,   eq. 113

where I is an (N+1)×(N+1) identity matrix,

$\begin{matrix}{{\begin{bmatrix}{{\partial U_{y}}/{\partial\left( z^{\prime} \right)}} \\{{\partial S_{x}}/{\partial\left( z^{\prime} \right)}}\end{bmatrix} = {\begin{bmatrix}0 & \left( {Einv}^{- 1} \right)_{reduced} \\B_{reduced} & 0\end{bmatrix}\begin{bmatrix}U_{y} \\S_{x}\end{bmatrix}}},} & {{eq}.\mspace{14mu} 114}\end{matrix}$

which reduces to

[∂² U _(y)/∂(z′)²]=[(Einv⁻¹)_(reduced) B _(reduced) []U _(y)].   eq. 115

The solution to eq. 115 is

$\begin{matrix}{U_{y\; i} = {\sum\limits_{m = 1}^{N + 1}{w_{i,m}\begin{Bmatrix}{{c_{m}^{+}{\exp \left( {{- k_{0}}q_{m}z} \right)}} +} \\{c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}\end{Bmatrix}}}} & {{eq}.\mspace{14mu} 116} \\{S_{xi} = {\sum\limits_{m = 1}^{N + 1}{v_{i,m}{\begin{Bmatrix}{{{- c_{m}^{+}}{\exp \left( {{- k_{0}}q_{m}z} \right)}} +} \\{c_{m}^{-}{\exp \left\lbrack {k_{0}{q_{m}\left( {z - d} \right)}} \right\rbrack}}\end{Bmatrix}.}}}} & {{eq}.\mspace{14mu} 117}\end{matrix}$

Here we first define Q as a diagonal matrix with elements q_(m), whichare the square roots of the N+1 eigenvalues of the matrix(Einv⁻¹)_(reduced)B_(reduced), and W as the (N+1)×(N+1) matrix formed bythe corresponding eigenvectors, with elements w_(i,m).

To find the matrix V, we can substitute eqs. 116 and 117 into the tophalf of eq. 114 to obtain

WQ=(Einv⁻¹)_(reduced) V,   eq. 118

so

V=[(Einv⁻¹)_(reduced)]⁻¹ WQ=Einv_(reduced) WQ.   eq. 119

Eq. 119 suggests a way to further improve the efficiency of thealgorithm for the TM case. A (2N+1)×(2N+1) matrix inversion is requiredto find the elements of Einv⁻¹, which is then reduced and inverted (an(N+1)×(N+1) matrix inversion) to find [(Einv⁻¹)_(reduced]) ⁻¹. But since[(Einv⁻¹)_(reduced)]⁻¹=Einv_(reduced), we may as well findEinv_(reduced) through application of eqs. 105 and 107 (with Einv inplace of Einv⁻¹), and invert that matrix to find (Einv⁻¹)_(reduced).This still involves an (N+1)×(N+1) matrix inversion, but eliminates the(2N+1)×(2N+1) matrix inversion, and we still end up with Einv_(reduced)as well as (Einv⁻¹)_(reduced) to use in eqs. 114 and 115.

In other words, instead of starting with the unreduced Einv⁻¹ matrix,start by forming Einv from the inverse permittivity components and apply

$\begin{matrix}{{\frac{\partial U_{y\; 0}}{\partial\left( z^{\prime} \right)} = {{({Einv})_{0,0}S_{x\; 0}} + {\underset{m = 1}{\overset{\infty}{\sum 2}}({Einv})_{0,m}S_{xm}}}},} & {{eq}.\mspace{14mu} 120} \\{i = 0} & \; \\{and} & \; \\{{\frac{\partial U_{y\; i}}{\partial\left( z^{\prime} \right)} = {{({Einv})_{i,0}S_{x\; 0}} + {\sum\limits_{m = 1}^{\infty}{\begin{bmatrix}{({Einv})_{i,m} +} \\({Einv})_{i,{- m}}\end{bmatrix}S_{xm}}}}},} & {{eq}.\mspace{14mu} 121} \\{i > 0} & \;\end{matrix}$

to find Einv_(reduced) use in eq. 119, and the inverse of this(N+1)×(N+1) matrix becomes (Einv⁻¹)_(reduced) for eq. 115. With thisenhancement and the use of eq. 112 for the reduced B matrix, thereremains only one (2N+1)×(2N+1) matrix operation to invert the originalpermittivity matrix, E, for the TM case. All other matrix operationsinvolve matrices of size N+1. Since matrix inversion scales as N³, Theelimination of one of the (2N+1)×(2N+1) inversions can have a largeimpact on computation time, especially for large truncation order N.

Eqs. 120 and 121 can be compared with eq. 42 from U.S. Pat. No.6,898,537, which again neglects inverse permittivity components wheni+j>N.

The above reductions for the TE and TM case can be applied to polarizedincident light, but can also be used with un-polarized light, by makinguse of eq. 1. This is particularly advantageous for obtaining below DUVreflectance data for the reasons already mentioned above. A typicalbelow DUV-Vis optical CD measurement would proceed as follows:

-   -   1) Pattern recognition and a separate vision system move the r-θ        stage to the desired grating structure.    -   2) Un-polarized light is directed on the structure from a normal        incidence configuration, and the specular reflection recorded.    -   3) A theoretical model of the assumed grating structure is        constructed.    -   4) A regression analysis is performed—either using real-time        model calculations or extracting the model curve from a library        database—to optimize the structural parameters of the        theoretical model, based on the measured reflectance. The above        reduced calculation for TE and TM polarized light, along with        eq. 1 may be used to perform the model calculations.

The optimized parameters at the end of the analysis are the measurementresult. The above steps, especially when used with a pre-generatedlibrary, can ordinarily be carried out in only a few seconds permeasurement. The particular configuration with normally incidentunpolarized light removes difficult issues such as polarizing below DUVradiation and alignment of the polarization to a particular direction,and is also easily integrated with an r-θ sample stage.

Aside from grating height and width, more complicated profile structurescan be measured by employing the recursive multiple layer RCW methodherein discussed, using a staircase approximation of the grating shape.In other words, the grating is sliced into a number of rectangularslices, each of different width, and the multiple layer RCW calculationemployed to compute the resulting diffraction efficiencies. The numberof slices used is chosen so that the calculated diffraction efficienciesconverge to the true diffraction efficiencies of the exact profileshape.

The parameters in the regression can be generalized and constrained, sothat a complicated profile shape can be modeled and optimized withouttesting unnecessary and unphysical situations. For example, atrapezoidal shape can be characterized by a top width, bottom width, andtotal height. The model in this case actually consists of a stack ofthin rectangular layers constrained to the trapezoidal shape, butotherwise forced to be consistent with the 3 parameters describing itsshape. Therefore, the regression optimization only considers the threeparameters describing the trapezoidal line shape. Even more complicatedgeometries can be approximated by stacking several such trapezoids ontop of each other. Further constraint can be applied to the regressionby requiring that the top width of the bottom trapezoid be the same asthe bottom width of the next trapezoid in the stack, and so on. If thegrating is symmetric with respect to rotations about the center of theridges or grooves, the above reductions can be employed. Otherwise, thefull RCW calculation may be used. In many cases, the real structure isapproximately symmetric, so the grating model can be accordinglyconstrained, even if the profile shape is complicated. The aboveconsiderations can also be easily extended to structures having morethan one transition per period and consist of more than just ridge andgroove regions. For example, the grating ridges may have a sidewallcoating.

With regard to eq. 1, in general, when measuring polarizing samples witha reflectometer eq. 1 can be used as long as the light incident on thesystem is unpolarized and the optical path itself does not impart anadditional polarization dependence on either the incident or reflectedlight. As already mentioned, depolarizers can be used to counter theeffects of polarizing optics or detection systems. Additionally, thereare methods for constructing optical systems, such as placing successivemirrors in orthogonal optical planes, so that the effective polarizationon the light is negligible, even when the individual optical componentsimpart some polarization dependence.

An alternate technique disclosed herein might augment existing opticaltechnologies operating with below DUV reflectometry data. One furthertechnique could incorporate the normal incidence un-polarized below DUVreflectometer herein described with optical technologies that provide alarger data set, but operate in other wavelength regimes. For example,polarized DUV-Vis reflectance data could be combined with un-polarizedVUV reflectance. The DUV-Vis reflectometer could operate at normal ornon-normal angle of incidence, or even at multiple angles of incidence.

A below DUV ellipsometer, operating in the range from around 150 nm-800nm or a DUV-Vis ellipsometer operating from about 200 nm-800 nm could becombined with the below DUV un-polarized reflectometer. The two datasetswill compliment each other, and in some situations provide moreinformation than either one dataset alone. The ellipsometer could befurther modified to operate at multiple polar and azimuthal angles ofincidence. Since the rigorous scattering methods can be used todetermine ellipsometric data as well as reflectance data, such acombination could provide further decoupling when determining structuralparameters of scattering surfaces.

Generally, the optical properties of the films involved in the patternedareas are characterized using similar, but un-patterned versions of thesame film stacks. In some cases, the scribe area between patternedregions of a semiconductor wafer have the same film structure as thepatterned features, except that they are not etched. If these areas arenot present, specific unpatterned film test structures can be providednear the patterned features. If the test structures or scribe areas areclose enough to the measured patterned areas, optical data from the twoareas can be simultaneously analyzed and common properties of the areasconstrained to be the same during the analysis. One particularlyconvenient way to implement this is through use of an imaging vacuumultraviolet reflectometer of the type described in U.S. Pat. No.7,067,818, since the reflectance data from the two areas can besimultaneously collected. Aside from simultaneously analyzing the data,the ratio of the reflectance data can also be advantageously used, sincethis ratio is independent of the incident intensity, thus removing theneed to calibrate absolute reflectance of the reflectometer.

It will be recognized that the techniques described herein are notlimited to a particular hardware embodiment of optical metrology toolsbut rather may be used in conjunction with a wide variety of types ofhardware. Thus, the hardware described herein will be recognized asmerely being exemplary. Further, it will be recognized that thetechniques described herein may be utilized with a wide variety of typesof computers, processors, computer systems, processing systems, etc.that may perform the various calculations provided herein in conjunctionwith collected data. Further, it will be recognized that the varioustechniques described herein may be implemented with software that mayreside on a computer or machine readable medium. For instance thevarious calculations described herein may be accomplished throughstandard programming techniques with computer programs that operate on acomputer, processor, computer system, processing system, etc.

Further modifications and alternative embodiments of this invention willbe apparent to those skilled in the art in view of this description. Itwill be recognized, therefore, that the present invention is not limitedby these example arrangements. Accordingly, this description is to beconstrued as illustrative only and is for the purpose of teaching thoseskilled in the art the manner of carrying out the invention. It is to beunderstood that the forms of the invention herein shown and describedare to be taken as the presently preferred embodiments. Various changesmay be made in the implementations and architectures. For example,equivalent elements may be substituted for those illustrated anddescribed herein, and certain features of the invention may be utilizedindependently of the use of other features, all as would be apparent toone skilled in the art after having the benefit of this description ofthe invention.

1. A method for measuring properties of a sample, comprising: providingan optical metrology tool that includes a first optical metrologyapparatus, the first optical metrology apparatus being a firstreflectometer having at least in part below deep ultra-violet lightwavelengths; and providing a second optical metrology apparatus withinthe optical metrology tool, the second optical metrology apparatusproviding optical measurements for the sample utilizing a differentoptical metrology technique as compared to the first optical metrologyapparatus, wherein data sets from the first optical metrology apparatusand the second optical metrology apparatus are combined and analyzed inorder to measure at least one property of the sample.
 2. The method ofclaim 1, wherein first reflectometer and the second optical metrologyapparatus operate at different wavelength ranges.
 3. The method of claim2, wherein the second optical metrology apparatus is an ellipsometerhaving at least in part deep ultra-violet or longer wavelengths.
 4. Themethod of claim 1, wherein the second optical metrology apparatus is apolarized second reflectometer operating at wavelength ranges at leastin part above vacuum ultra-violet wavelengths.
 5. The method of claim 1,wherein the second optical metrology apparatus is a polarizedreflectometer operating at wavelengths that include at least in partwavelengths below deep ultra-violet wavelengths.
 6. The method of claim1, wherein the first reflectometer is configured for normal incidence.7. The method of claim 6, further comprising using a reduced RCWcalculation for analyzing 2-D periodic structure of the sample.
 8. Themethod of claim 6, further comprising using a group theoretic approachfor analyzing 3-D periodic structure of the sample.
 9. The method ofclaim 6, wherein incident light of the first reflectometer isun-polarized.
 10. The method of claim 9, wherein the first reflectometeremploys an r-θ stage.
 11. The method of claim 10, wherein a calculatedreflectance of the sample is obtained from a relationshipR=0.5*(R_(TE)+R_(TM)), regardless of sample rotation.